3 the physics of cathode processessst/teaching/ame60637/reading/2008_anders...76 3 the physics of...

3

The Physics of Cathode Processes

About thirty years ago – I remember it very well – I wonderedabout the incompatibility of model ideas and experimental facts.What does really happen in the cathode arc spot? At that time,I imagined the great magma lake of the Halemaumau crater inHawaii, with its red-hot, glowing, molten surface, steaming,boiling, bubbling, gushing, in restless motion, flowing andwhirling, rising and falling, covered by clouds of hot vapours, andoften throwing gigantic, bright fountains of liquid magma,coupled with ejection of large showers of glowing droplets andpieces and lumps of magma. – So, similar, as it seemed to me,must be the surface of an arc spot, if we could observe it in action,in quite different time and space scales, of course.

Erhard Hantzsche, in his Dyke Award Address, XX ISDEIV,Tours, France, 2002

The fractal approach is both effective and ‘‘natural.’’ Not onlyshould it not be resisted, but one ought to wonder how one couldhave gone so long without it.

Benoit Mandelbrot, in ‘‘The Fractal Geometry of Nature,’’1982

Abstract This chapter is at the heart of the book. It is the longest chapter that

deals with electron emission processes. The basic mechanisms of electron emis-

sion are outlined, including thermionic emission, field emission, their nonlinear

combination, as well as explosive emission. This leads to the non-stationary

emission centers, which are the sources of electrons and plasma. The statistical

nature of emission center ignition, coupled with self-similar features of emission

centers in space and time, lead naturally to a description of cathode spots as a

fractal phenomenon. If taken seriously, the old discussion of the ‘‘true’’ current

density and ‘‘true’’ characteristic time of cathode spots needs to be re-evaluated:

those properties are fractal down to the physical cutoffs, which are generally still

below the resolution limits of the experimental equipment. As a consequence,

A. Anders, Cathodic Arcs, DOI: 10.1007/978-0-387-79108-1_3,� Springer ScienceþBusiness Media, LLC 2008

75

some new features of cathode phenomena are introduced, such as the existenceof transient holes in the cathode sheath.

3.1 Introduction

Cathode processes are at the heart of cathodic arc plasma production. Cathodicarc processes have features that are surprising or strange, even for scientists andengineers familiar with discharges and plasmas. The plasma ‘‘root’’ or ‘‘attach-ment’’ on the cathode is localized in bright, tiny spots that appear to quicklymove more or less randomly on the cathode surface. In the presence of anexternal magnetic field, spot motion appears to be steered in a preferred (closeto the ‘‘anti-Amperian,’’ –j�B) direction. In any case, the cathode materialsuffers a transition to dense plasma at these spots, and the dense cathodeplasma expands rapidly into the vacuum or low-pressure gas ambient. Theanode is merely a passive collector of electrons unless its area is very small or itis thermally isolated or the discharge current is very high. In these cases, anodespots may appear and/or the anode may evaporate. Because this book focuseson cathodic arc deposition, these special anode conditions are not muchconsidered.

The structure and dynamics of cathode spots are still the subject of dis-cussions and research. The current density plays a central role in manymodels because it is critically important for power density distributionand energy balance, which, in turn, govern all processes of electron emission,phase transitions, and plasma production. Most, but not all, experts agreethat explosive electron emission and the associated ‘‘ecton’’ model is closestto reality. An emerging model is the concept of a stochastic fractalmodel for the cathode spot, which includes ecton processes as the determin-ing lower limit of self-similar scaling. The fractal model may be suitable todescribe what Erhard Hantzsche expressed in his Dyke Award Address(see the quote at the beginning of this chapter). Throughout this and otherchapters, the fractal model approach will be described to the extent known atthis time.

The first question that may arise could be formulated as follows: Why is thecathode active but the anode passive? This question touches the general role ofcathode and anode in gas discharges. It is clear that electric current in bothcathode and anode is associated with electron motion in the conduction band ofthe metals (solid or liquid) that make up cathode and anode. If plasma happensto be present between cathode and anode, current can flow bymotion of free andmobile charged particles. In the plasma, most of the electric current is carried byelectrons because electron mobility is much higher than the mobility of ions, aconsequence of their very different masses.

The critical places of current continuity are the interfaces between plasma andmetal. Electrons move to the anode and fall there into the anode’s conduction

76 3 The Physics of Cathode Processes

band, liberating the potential energy known as the work function (about 4 eVper electron for most metals), which leads to heating of the anode. Currentcontinuity (Kirchhoff ’s law, 1846) is therefore easily accomplished at the anode.A small anode potential drop usually appears to adjust the arrival rate ofelectrons at the anode surface to the right current as required by currentcontinuity.

In contrast, electrons in the conduction band of the cathode are preventedfrom escaping by a potential barrier, the work function of the cathode. Elec-trons need to be given the work function energy to ‘‘free’’ them from thecathode. Here is where the nature of the discharge creates special conditionsenabling electrons in the cathode to overcome the potential barrier that keepsthem normally inside the cathode. Depending on the character of those‘‘special conditions,’’ we distinguish different electron emission mechanisms,and they ultimately lead to different forms of electrical discharges. Goinga step further one can define the type of discharge by the electron emissionmechanism.

To understand and appreciate cathodic arc operation, a brief overview ofthese ‘‘special conditions’’ leading to electron emission needs to be given. As wewill see, electrons can be emitted by individual events, such as ion impact or bycollective effects affecting all electrons in the cathode, such as high cathodetemperature and/or high electric field on the cathode’s surface. Following pio-neering work of Hantzsche [1, 2], one can use the distinction between individualand collective emission processes to define glow and arc discharges in a verygeneral, physical manner, regardless of phenomenological appearance or abso-lute values of current and voltage.

The potential distribution between cathode and anode is generically andschematically shown in Figure 3.1. The voltage is not evenly distributed but

Fig. 3.1. Schematic, generic, one-dimensional presentation of the potential distributionbetween cathode and anode; the sheaths are actually very thin and shown not to scale.One should note that cathode-spot formation, as explained later, requires modeling of a

two or better three-dimensional potential distribution near the cathode surface, not thesimplified one shown here

3.1 Introduction 77

concentrated in the sheaths near the cathode and anode. The potential drop inthe cathode sheath is most significant for the mechanism that liberates electronsfrom the cathode. The anode drop, in contrast, can be positive or negativedepending on arc current, anode area, and other factors affecting the electroncurrent arriving at the anode.

Themagnitude of the cathode fall is a ‘‘fingerprint’’ of the cathodemechanism.The cathode mechanism adjusts the cathode fall (i.e., the voltage drop in cathodesheath) that is needed to maintain the discharge. In glow discharges, ions comingfrom the bulk plasma are accelerated toward the cathode when they enter thecathode sheath, and thus they arrive at the cathode surface with approximatelythe energy (in eV) gained in the cathode fall (expressed in V), if we assume thatthey are typically singly charged and did not suffer collisions. The yield ofsecondary electron emission is greatly affected by the potential energy of theimpacting ion (potential emission, discussed in greater detail in Chapter 8). Thevoltage drop in a glow discharge usually exceeds 300V and is often in the regionof 400–500V in order to give secondary electrons enough energy to heat plasmaelectrons and to cause ionization in the plasma bulk. The cathode fall is self-adjusting as one can see by the following argument. If electrons do not gainenough energy in the sheath, the plasma generation will be reduced, the plasmadensity starts to drops, and the plasma impedance (complex resistance) increases.This will cause an increase in the burning voltage, which also means that thevoltage drop in the sheath increases, thereby leading to an increase in secondaryelectron energy, which is followed by an increase in plasma density.

Secondary electrons are vital for the operation of glow discharges. In arcs, incontrast, electrons are not released by individual ion impact but by a collectivemechanism, which can be thermionic (i.e., the cathode is at very high tempera-ture) or determined by a strong electric field. The hot cathode case leads tothermionic arcs, which can be stationary. In some situations, a hot cathode isalso subject to a high electric field. Nonlinear amplification of thermionic andfield emission can occur, known as thermo-field emission.

Thermo-field emission can be coupled to a thermal runaway process, namelywhen emission is related to energy dissipation and net heating of the cathode,which in turn can enhance the temperature and associated electron emission.Locations where this occurs can explosively evaporate, leading to a new form ofelectron emission that is inherently non-stationary because the emission locationis changed by the explosion, subsequent plasma expansion, and the increase ofthe hot spot area by thermal conduction. This non-stationary form of emission iscalled explosive electron emission. The explosive phase requires a minimumaction to be invested, a ‘‘quantum’’ of the explosive process, the so-calledecton. The ecton phase is generally much shorter than the overall cycle or ‘‘life-time’’ of an emission center. Ignition of a competing emission center is closelyrelated to the existence of the current center. The dense plasma provides theconditions for the ignition while ‘‘choking’’ the already operating emissioncenter by its limited conductivity. These features will be discussed later in thechapter.


Ignition of emission center appears to be a very important element to under-

standing cathodic arcs. Ignition in this sense is not just the triggering of the arc

discharge but the arc’s perpetual mechanism to ‘‘stay alive.’’ The probabilistic

distribution of perpetual ignition of emission centers can be associated with a

fractal spot model [3].In the remainder of this chapter, electron emission theory will be briefly

described as it is related to arcs, before explosive emission, crater formation,

metal plasma generation, ectons, the fractal features of the cathodic arc spots,

spot types, and arc modes are discussed in greater detail.

3.2 Theory of Collective Electron Emission Processes:

Steady-State Models

3.2.1 Thermionic Emission

The theory of electron emission has been dealt with in many publications and

textbooks. Therefore only the essential results are reproduced here. They are the

foundation for the development of a theory of cathodic arc operation.Electrons in the conduction band of a metal can be described by the model

of a free electron gas. At the beginning of the twentieth century, Boltzmann

statistics was applied to free electrons (Drude model), and some but not all

properties could be explained. Later it was recognized that electrons, spin ½

particles, are subject to Fermi–Dirac quantum statistics. The expression

fF E;Tð Þ ¼ 1

1þ exp E�EF

kT

� � (3:1)

is the Fermi distribution function which describes the probability that a state of

energy E will be occupied in thermal equilibrium at a temperature T. Figure 3.2

Fig. 3.2. Fermi distribution function of electrons in metal at zero and elevated tempera-tures. This example illustrates the situation for copper: even at the melting temperature,only a small fraction of electrons is thermally excited and smears out the region around

the Fermi energy, EF (see [4] p. 15)

3.2 Theory of Collective Electron Emission Processes: Steady-State Models 79

illustrates this function using the example EF � 7eV, which is approximately

applicable to copper. The Fermi energy, EF, can be defined as the highest energy

of occupied states at temperature T ¼ 0; it depends only on the density:

EF ¼�h2

2me

� �3p2n� �2=3

: (3:2)

In the above equation, �h ¼ h=2p, h is the Planck constant, k is the Boltzmann

constant, andme is the electron mass. The Fermi distribution implies that only a

small fraction of electrons can acquire energy of order kT above the Fermi

energy. From Figure 3.2 one can see that even at the boiling temperature of

copper, T ¼ 1; 356K, most of the electrons are still part of the rectangular

distribution and not influenced by the high temperature. This feature will be

important later when discussing the Nottingham effect.The expression

dN E;Tð ÞdE

¼ 4p 2með Þ3=2

h3E1=2

1þ exp E�EF

kT

� � (3:3)

is called the Fermi–Dirac distribution and describes the number of electrons

with a kinetic energy in the range (E, E+dE). The shape of the Fermi–Dirac

distribution is shown in Figure 3.3, left. Often, for convenience of energy con-

siderations, the same function is presented with the abscissa and ordinate

exchanged (Figure 3.3, right).The electrons of the ‘‘electron gas’’ are confined to the solid volume by a

potential barrier, �, above the Fermi level (Figure 3.4). � is generally known as

the work function and expressed in eV (electron-volt1). Data on work functions are

available from many experiments [5] and from quantum-mechanical modeling [6].An intuitively clear picture is to heat the electron gas so that some electrons

will have enough energy to overcome the barrier. The classic condition for

escape would be simply

Ez � �; (3:4)

where Ez ¼ mev2z

�2 is the kinetic energy of an electron in the z-direction, the

direction perpendicular to the surface. The emission current density is

jthermionic ¼ en Eð Þvz; (3:5)

where n Eð Þ is the density of electrons satisfying the condition (3.4). For T ¼ 0

obviously no electrons satisfy (3.4). For elevated temperature one can calculate

n Eð Þ by integrating the distribution function from �1 to þ1 in the x- and

y-directions and from � to1 in the z-direction. Here one can use a convenient

simplification. At elevated temperature, the high-energy tail of the Fermi

1 In some context, the barrier is expressed as a potential, with the unit volts; the factor ofproportionality is simply the elementary charge, e.


distribution (3.1) is dominated by the exponential term, where E� EFð Þ4kT,

and the quantum-statistical Fermi distribution can be well approximated by the

classic Boltzmann distribution

fB E;Tð Þ ¼ exp �E� EF

kT

� �: (3:6)

The result of the integration, using the Boltzmann distribution, is

jthermionic ¼ AT 2 exp � �

kT

� �; (3:7)

where

A ¼ 4pemek2

h3¼ 1:202� 106 A=m2 K2: (3:8)

Fig. 3.3. Left: The Fermi–Dirac distribution (3.3). At temperature greater than zero,

energy states higher than the Fermi energy are occupied, and some states lower than theFermi energy are not occupied. Right: Presentation of the Fermi–Dirac distribution withabscissa and ordinate exchanged, as it is common for energy considerations. The energy

levels indicated below are electrons not part of the electron gas but bound to a specificlattice site. The energy level of free electrons outside the solid is indicated by vacuum levelat the very top of the energy scale. The difference � between the Fermi level and the

vacuum level is called the work function, as discussed in the text


Equation (3.7) is the well-known Richardson–Dushman equation for thermio-

nic emission,2 and A is the universal Richardson constant.In the above derivation it was assumed that the potential barrier is uniform

throughout the surface, and hence effects of the atomic modulation, crystal-

lographic structure, and direction were neglected. Additionally, the likely pre-

sence of oxide or water layers, adsorbates, and surface defects was not taken into

account. Because of the simplifications, it should not surprise that experimental

values for the constant A vary widely. The relation (3.7) has been verified many

times, but usually the data are fitted by adjusting both the work function � and

the Richardson constant A. Representative data for � and A are compiled in

Table B2 (Appendix B).

3.2.2 Field-Enhanced Thermionic Emission

In the previous section, any effects of a possible electric field were not consid-

ered. In practically all cases, however, an electric field is present, namely the field

caused by the emitted electron and external fields. A free electron located at a

distance z from the surface outside the solid causes a rearrangement of electrons

of the electron gas in the solid in such a way that the resulting charge distribution

in the solid looks like a mirror image of the free charge. The free electron and its

‘‘image’’ experience the Coulomb force

Fig. 3.4. Potential barrier for electrons at the surface of a metal. The barrier is deformedand its height reduced due to the mirror image field and the applied external electric field.

The reduced barrier height, �S, allows more electrons to classically leave the metal at hightemperature (Schottky effect)

2 Owen Williams Richardson [7], when still a graduate student at Trinity College, made

an almost correct derivation, read before the Cambridge Philosophical Society in 1901,i.e., at a time before quantum statistics was known. He later published a book on thesubject [8]. Saul Dushman made improvements in the 1920s [9] and wrote a compre-hensive review [10].


F ¼ e2

4p"0 2zð Þ2(3:9)

and their interaction potential is

W ¼ e2

16p"0z: (3:10)

An external electric field can be written in linear approximation near the surface

since it will slowly vary over the distance of interest. The potential distribution

perpendicular to the surface can hence be written as

V zð Þ ¼ �� e2

16p"0 z� eEz; (3:11)

where the symbol E is used for the external electric field near the surface.3 The

maximum of the potential barrier can be determined by dV=dz ¼ 0, which gives

the location for the maximum of the potential barrier

zmax ¼e

16p"0E

� �1=2

(3:12)

from the surface. Putting (3.12) in (3.11) results in the maximum height of the

potential barrier

�S ¼ ��e3E

4p"0

� �1=2

; (3:13)

which is illustrated in Figure 3.4. One can interpret this result that the barrier

height is reduced compared to the usual work function, which of course leads to

a higher emission current (Schottky effect). The current density with Schottky

effect can therefore be written as

jthermionic;Schottky ¼ AT2 exp � �SkT

� �: (3:14)

One needs to note that the correction term e3E�4p"0

� �1=2needs to be small

compared to the work function. If this is not the case, the general expression of

thermo-field emission needs to be applied, as discussed in Section 3.2.4.Up to now, the emission of electrons has been dealt with classically. Only the

appearance of the Planck constant, h, in the Richardson constant A, (3.8),

indicated that quantum effects may be involved. So far, the quantum nature of

the electron gas was mentioned but the emission process was entirely classical

since electrons were given energy to overcome the potential barrier. The situation

changes when the electric field is very strong, which is described in the next section.

3 The symbol E is here used for the electric field to avoid confusion with E, the symbolfor energy.


3.2.3 Field Emission

At very high field strength, the potential barrier changes its shape to become a

hill of sufficiently small width through which electrons can tunnel quantum-

mechanically (Figure 3.5). Tunneling through the barrier can even occur at

temperatureT ¼ 0. A general approach to calculate the emission current density

is to write

je ¼ e

ð1

�eVmin

N Ez;Tð ÞD Ez;Eð Þ dEz; (3:15)

whereN Ez;Tð Þ is the electron supply function that essentially contains the Fermi

distribution (3.1) with the variable Ez ¼ mev2z

�2, the kinetic energy of electrons

in the z-direction, which is defined as perpendicular to the surface, andD Ez;Eð Þis the tunneling probability through the potential hump. The integration covers

all electron energies beginning from the lowest energy of electrons of the electron

gas in the metal. To calculate the tunneling probability, one solves Schrodinger’s

wave equation in the three regions: inside the metal, inside the barrier, and

outside themetal and barrier. In the limiting case of low temperature one obtains

the Fowler–Nordheim formula [11]

jFN Eð Þ ¼ e3

8ph � t2 yð ÞE2 exp � 8p

ffiffiffiffiffiffiffiffiffiffiffiffiffi2me�3

p

3ehEv yð Þ

" #

; (3:16)

where

y �

ffiffiffiffiffiffiffiffiffiffie3E

4p"0

s1

�(3:17)

Fig. 3.5. Potential barrier at the surface of a metal: at very high electric field strength, its

shape becomes a hill of sufficiently small width through which electrons can tunnelquantum-mechanically


and t yð Þ and v yð Þ are elliptical functions whose numerical values [12–14] are

given in Table 3.1. A convenient numerical form of (3.16) is

jFN Eð Þ ¼ 1:541� 10�6E

2

� t2 yð Þ exp �6:831� 109�3=2v yð Þ

E

� (3:18)

with

y ¼ 3:795� 10�5ffiffiffiffiEp

�(3:19)

using E in V/m, � in eV, and jFN

in A/m2. One should note that the function t yð Þhas only a marginal influence whereas v yð Þ greatly affects the calculated current

Table 3.1. Numerical values of elliptical functions

used in Fowler–Nordheim formula

y t(y) V(y)

0.00 1.000 1.000

0.05 1.001 0.9950.10 1.004 0.9820.15 1.007 0.962

0.20 1.011 0.9370.25 1.016 0.9070.30 1.021 0.8720.35 1.026 0.832

0.40 1.032 0.7890.45 1.038 0.7410.50 1.044 0.690

0.55 1.050 0.6350.60 1.057 0.5770.65 1.063 0.515

0.70 1.070 0.4500.75 1.076 0.3830.80 1.083 0.312

0.85 1.090 0.2380.90 1.097 0.1610.95 1.104 0.0821.00 1.110 0.000

1.1 1.125 –0.1721.2 1.139 –0.3531.3 1.153 –0.545

1.4 1.167 –0.7451.5 1.180 –0.9542.0 1.249 –2.122

3.0 1.380 –5.2134.0 1.501 –8.4585.0 1.615 –12.438


density. Therefore, v yð Þ requires careful treatment. Hantzsche [15] provided theconvenient analytical approximation

v yð Þ ffi 1� y2 0:375 ln 1=yð Þ þ 0:9673½ � � 0:0327y4 for y �1�1:3110y3=2 þ 0:8986y1=2 þ 0:4936y�1=2 � 0:0812y�3=2 for y 1.

(3:20)

3.2.4 Thermo-field Emission

Up to now we have emission formulas for two limiting cases. At very hightemperature and relatively low electric field one would use the Richardson–Dushman formula with Schottky correction for the work function, see (3.14).The other extreme case is for very high electric fields but low temperature: hereone would use the Fowler–Nordheim formula (3.16). In many practical cases,however, especially when considering arc discharges, both temperature andelectric field strength are high and therefore the general approach (3.15) has tobe solved without the assumption of low temperature. The resulting theory is notsimply a superposition of classic thermionic emission and quantum-mechanicalfield emission but the quantum-mechanical solution of (3.15). Pioneering workwas done by Dolan and Dyke [16], andMurphy and Good [17] in the 1950s. Thegeneral expression for the current density is somewhat cumbersome:

jTFðT;E; �Þ ¼4pmekT

h3

ðWl

�WA

ln 1þ exp � Ezþ�kT

� ��

1þ exp8p 2með Þ1=2v yð ÞE3=2

z

3he

h idEz:

þð1

Wl

ln 1þ exp �Ez þ �kT

� �dEz

�

8>>>>>>>><

>>>>>>>>:

9>>>>>>>>=

>>>>>>>>;

; (3:21)

where y was defined in (3.17),

vðyÞ ¼ �ffiffiy2

q�2E k1ð Þ þ 1þ yð ÞK k1ð Þ½ � for y �1

ffiffiffiffiffiffiffiffiffiffiffi1þ yp

E k2ð Þ � y K k2ð Þ½ � for y 1

8<

:; (3:22)

E kð Þ ¼ðp=2

0

1� k2 sin2 �� 1=2

d�; (3:23)

K kð Þ ¼ðp=2

0

1� k2 sin2 �� 1=2

d�; (3:24)


k1 ¼y� 1

2y

� �1=2

and k2 ¼1� y

1þ y

� �1=2

: (3:25)

In the integral boundary,WA is the height of the potential barrier with respect tothe lowest energy of free electrons in the metal (� was the barrier height relativeto the Fermi level) and

Wl � �e3E

8p"0

� �1=2

: (3:26)

Because these formula are quite involved, simplified formula are often used thatare good approximations for certain temperature and field regions. For exam-ple, Christov [18] developed a general expression with three relatively simpleintegrals. Hantzsche [15] developed a number of additive and harmonic combi-nations of thermionic and field emission formula such as

jTF T;Eð Þ � k AT2 þ BE9=8 �

exp � T2

Cþ E

2

D

!�1=22

4

3

5; (3:27)

where A ¼ 120; B ¼ 406; C ¼ 2:727� 109; D ¼ 4:252� 1017; the units for(3.27) are jTF in A/cm2, T in K, E in V/cm, and � � 4:5 eV. The constants A,B, C, and D are fitted such as to minimize deviation from the more accurateformula (3.21). Although the expressions (3.21)–(3.25) can be easily pro-grammed with today’s computers and software, the use of fit formula (3.27)may still be valuable if computational speed matters.

Coulombe and Meunier [19] compared emission current densities calculatedby the Richardson–Dushman equation (including Schottky correction) withcurrent densities by Murphy and Good. It was shown that the Richardson–Dushman equation always gives lower values than the more accurate treatmentby Murphy and Good.

3.3 Refinements to the Electric Properties of Metal Surfaces

3.3.1 Jellium Model and Work Function

A simple model for the electronic properties of a metal surface is called thejellium model [20, 21]. In this model, the atoms of the solid metal are described aspositive ion cores in a sea of free electrons; the ion cores are seen as smeared outto produce a uniform positive charge.

The electronic charge density at the surface does not drop in a mathemati-cally sharp manner but exponentially. The electrons accumulated on the outeredge leave a positively charged region behind, thus forming a dipole layer(Figure 3.6). The potential related to this dipole is the surface space chargeor dipole potential, Vdipole zð Þ.

3.3 Refinements to the Electric Properties of Metal Surfaces 87

The electron contribution of the electric potential can be divided into threecontributions [21]:

V zð Þ ¼ Vcore zð Þ þ Vexchange zð Þ þ Vdipole zð Þ: (3:28)

The term Vcore zð Þ describes the potential between core electrons and valenceelectrons; Vcore zð Þ is practically independent on whether the atom is in the bulkor on the surface of the solid because of the localized nature of the core electrons.Therefore, Vcore zð Þ cannot significantly contribute to surface effects such as theformation of the potential barrier (work function). Vexchange zð Þ is the exchangepotential between valence electrons, also known as exchange–correlation poten-tial. Electrons lower their energy by ‘‘avoiding’’ other electrons of like spin (Pauliexclusion principle) and due to Coulomb interaction (correlation interaction).As a result, a deficit of electronic charge is around each electron, which, togetherwith the dipole potential, is responsible for keeping electrons inside the metal.

In one approach, one can solve the one-dimensional Poisson equation

d 2V zð Þdz2

¼ � 1

""0� zð Þ; (3:29)

where � zð Þ is the electric charge density, " is the dielectric constant in the solid,and "0 is the permittivity of free space. One can define the boundary conditionsV lDð Þ ¼ 0 and V 0ð Þ ¼ Vs, where lD is the distance from the surface into thesolid where the electron concentration attains bulk value and the electrostaticpotential becomes zero; the potential height on the surface is designated as Vs.The solution shows that [21]

lD ¼2""0Vs

enbulke

� �1=2

(3:30)

is the characteristic screening length, usually called the Debye length. Later,when discussing arc plasmas, a similar Debye length is introduced, where thepotential energy eVs is replaced by the thermal kinetic energy kT of chargedplasma particles. The solution V zð Þ is shown in Figure 3.6. It shows damped

Fig. 3.6. Distribution of the electronic charge density at the surface of a metal


oscillations penetrating the solid to some depth. They are sometimes called

Friedel oscillations. For typical electron concentrations of a metal, about

1028 m–3, the depth defined by (3.30) is extended to only about one atomic

layer because free electrons of the metal screen the field. For semiconductors

and insulators, the electron concentration is much smaller and therefore the field

may penetrate thousands of atomic layers into the bulk.In another approach of treating the jellium model [20], the wave function of

the whole system of electrons can be written as a superposition of one-electron

Schrodinger wave functions

� �h2

2me

� �r2 i þ V i ¼ Ei i; (3:31)

where Ei is the total allowed electron energy. If one supposes that the solid is

infinite in x- and y-directions and has a surface at z ¼ 0 and it can be divided into

boxes (cubes) of side length L, the solution are the eigenvalues of the wave

equation:

k ¼2

L3

� �1=2

sin kzzð Þ exp i kxxþ kyy� ��

; (3:32)

where k is the electron wave vector, with the allowed values kz ¼ Np=L,N ¼ 1;2;3;:::. The charge density is related to the wave function by the general

relation

� ¼ �eX

k

kj j2 (3:33)

from which one also obtains damped oscillations of the charge density from the

surface into the solid. In this approach, the Fermi length

lF ¼2pkF¼ 8p

3nbulke

� �1=3

(3:34)

is an often-used scaling length, which is typically 0.5 nm for metals, and

kF ¼ 3p2nbulke

� �1=3(3:35)

is the Fermi momentum corresponding to the momentum of electrons at the

highest energy at T ¼ 0, the Fermi energy (3.2).At the beginning of this section, the work function was simply introduced as

the potential barrier height for the most energetic electrons at T ¼ 0, i.e., for

electrons at the Fermi energy. Now, with a deeper understanding of the electro-

nic structure of the surface, one can write

�=e ¼ Vexchange þ Vdipole � VFermi; (3:36)

where VFermi ¼ EF=e is the Fermi potential. If we go beyond the jellium model

and consider the crystalline structure of metals, it is not difficult to comprehend

that the surface dipole potential, Vdipole, depends on the distance and charge of

the positive ion cores. Therefore, different crystalline surfaces, even of the same


element, exhibit differences in the work function, as was experimentally con-

firmed; see, e.g., [22].

3.3.2 The Role of Adsorbates

In all of the previous sections, it was assumed that the surface of the metal is

uniform and free of any defects or adsorbed layers. These assumptions were

useful and necessary to derive a fundamental understanding of the emission

processes. The reality, however, is much more complicated. In fact, the idealized

situation almost never applies, except in cases of carefully prepared surfaces in

ultrahigh vacuum. Cathode surfaces in typical deposition systems are usually

covered with non-metallic atoms and films. As will be discussed, adsorbates and

roughness affect the work function and the electric surface field. Since the work

function appears in the exponential terms of the emission formula derived ear-

lier, careful consideration is necessary. First it will be shown how quickly

cathode surfaces are covered with adsorbates. After the presence of adsorbates

has been demonstrated, their effect on the work function will be explained.Any surface is subject to impingement of the atoms or molecules4 of the gas

the surface is exposed to. From the kinetic theory of gases [20, 23, 24] we have the

impingement rate

Jg ¼p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pmgkTg

p ; (3:37)

where p, mg, and Tg are the pressure, mass, and temperature of the gas, respec-

tively. Even at high vacuum with a typical pressure of 10–4 Pa, the impinge-

ment rate cannot be neglected. For a system operating with background gases,

typical for reactive film deposition, the rates are very high (Table 3.2).

Table 3.2. Pressure (in Pascal and Torr), impingement rate, and monolayer formationtime for selected vacuum and process conditions

p (Pa) p (Torr) Jg (m–2 s–1) t (s)

Nitrogen

1 7.5 � 10–3 2.9 � 1022 3.5 � 10–4

10–1 7.5 � 10–4 2.9 � 1021 3.5 � 10–3

10–2 7.5 � 10–5 2.9 � 1020 3.5 � 10–2

water vapor10–3 7.5 � 10–6 3.6 � 1019 0.2810–4 7.5 � 10–7 3.6 � 1018 2.8

10–5 7.5 � 10–8 3.6 � 1017 28

4 The terms ‘‘atoms’’ and ‘‘molecules’’ are used synonymously in this chapter.


It is interesting to consider how long it would take to cover an initially clean

metal surface with a layer of gas molecules. For the purpose of this exercise one

could assume that all arriving gas molecules stick to the surface. With this

assumption, the monolayer formation time can be estimated by

tML ¼ ��Jg; (3:38)

where � � 1019m�2 is the areal atom density of the metal surface. Typical results

are included in Table 3.2. One can see that if a metal surface was initially

chemically clean, i.e., free of adsorbates, this feature can be maintained for

minutes or hours only under ultrahigh vacuum (UHV) conditions.The above assumption that all gas molecules stick to the clean metal surface is

of course over-simplified. When a gas molecule approaches the surface it inter-

acts with the atoms of the surface experiencing attractive and repulsive forces. In

most cases, the superposition of the attractive and repulsive potentials shows a

minimum at a small distance r0 from the surface, which can lead to trapping

(bonding) of the arriving atom (Figure 3.7). The nature of the attractive forces

determines the depth of the potential minimum �Va50 (to be discussed in

greater detail in the chapter on film growth). The trapped atom has a probability

of escape that can be expressed in an Arrhenius form [20] as

vdes ¼ �0 exp ��G�deskTs

� �; (3:39)

where �0 � 1013s�1 is called the attempt frequency, which is associated with the

vibration frequency of surface atoms,�G�des is the free energy of activation of the

Fig. 3.7. Schematic potential diagram for atom interacting with surface atoms; z is thedistance from the surface, r0 indicates the minimum of potential energy corresponding

to the equilibrium distance of the atom becoming trapped. The repulsive term ismainly due to overlap of filled electron orbitals of surface atoms with orbitals of thearriving gas atom. The attractive term depends greatly on the specific nature of theinteraction (Coulomb, covalent, polar, van der Waals)


desorption process, and Ts is the temperature of surface atoms. The desorptionprobability can also be expressed through the enthalpy �Hdes ¼ �e�Va via

vdes ¼ �0f �

fexp ��Hdes

kTs

� �; (3:40)

where f and f � are the molecular partition functions of the system in theequilibrium and activated states, respectively [20]. The reciprocal is the meansurface lifetime of the atom in the trapped state,

ta ¼ t0 exp�Hdes

kTs

� �; (3:41)

with

t0 ¼f

f �1

�0: (3:42)

Values of the adsorption energy, or energy necessary to desorb the adsorbedatom, �Hdes, varies greatly, as does the mean surface lifetime, ta. Depending onthe depth of the potential minimum, and thus the strength of bonding, onecustomarily distinguishes between weak physical adsorption (or physisorption)and much stronger chemical adsorption (or chemisorption). van derWaals forcesare typical for physisorption. Hydrogen bonding, covalent chemical bonding, andmetal bonding are typical for chemisorption. The transition is customarily set toabout 0.2 eV/atom, as the physicist would say, or 5 kcal/mol or 21kJ/mol, as thechemist sees it. Table 3.3 shows examples of physisorbed and chemisorbed gaseson cathode surfaces. As is clear from the exponential factor in (3.41), the ratio�Hdes=kTs is critical. Obviously, heating the cathode makes physisorption extre-mely short-lived; however, chemisorbed atoms may be difficult or impossible toremove evenwhen themetal approaches itsmelting temperature.We can concludethat a real surface is a highly dynamic object on which atoms frequently adsorb

Table 3.3. Examples of the mean surface lifetime, ta, of physisorped and chemisorped

gases on cathode surfaces [21], assumingTs ¼ 300 K (equivalent to kTs ¼ 0:0258 eV) and

t0 � 10�13s. The point of these examples is not the exact data but to demonstrate the hugeeffect of �Hdes

ExampleApproximate�Hdes (eV/atom)

Approximate ta(s)

H2 physisorped on metal 0.07 1.3 � 10–12

Ar, CO, N2, CO2 physisorped on metal 0.15–0.18 10–10

Carbohydrates physisorped or weaklychemisorped

0.4–0.6 10–6–10–2

H2 chemisorped on metal 0.9 100

CO on Ni 1.3 4 � 109

O on W 6.5 101100 4age ofthe universe


and desorb but also a more or less permanent non-metallic layer may have

formed. In any case, the presence of adsorbates is the rule, not the exception.The presence of adsorbates changes the work function. Figure 3.8 shows two

situations when an atom approaches the surface and the electronic charge of the

solid starts to overlap with the orbitals of the atom, i.e., adsorption occurs, and

thus full or partial charge transfer between the adatom and the metal will occur.

On the left side, the adatom has a filled electronic state slightly above the Fermi

level of the metal. In this case, electron transfer from the atom to the metal will

occur, and the adsorbed atom will assume a net positive charge, which will cause

a reduction of the work function. This becomes plausible if one considers that

the work function was associated with a dipole potential, (3.36), where the

negative charge was sticking out from the surface. The right side of Figure 3.8

shows the other case: the adsorbed atom has an unfilled state below the Fermi

energy, and therefore charge transfer to the adatom will occur, giving it a net

negative charge, which will enhance the work function.If a polar molecule is adsorbed, a similar phenomenon can occur even without

full electron transfer to the metal. If the polarized molecule is adsorbed with the

positive side of the dipole pointing away from the surface, the work function will

be reduced, and in the opposite case the work function will be enhanced.Inert gas, like argon, is polarizable andwill therefore ‘‘feel’’ the dipole field of the

surface. Adsorption of noble gases will change the charge distribution of adsor-

bed noble gases slightly in such a way that the work function is slightly lowered

([21] p. 369). Usually, chemisorption of hydrocarbons on transition metals also

reduces the work function by 1.2–1.4 eV [21]. On the other hand, adsorption of

hydrogen and oxygen atoms usually increases the work function of metals.The formation of an oxide layer (or more general, insulating layer) of several

atoms thickness prevents charge transfer to the metal in the case of an ion

arriving at the surface. If the kinetic energy of the ion is low, it may adsorb to

Fig. 3.8. Charge transfer from and to adatoms on a metal surface. Left: The adatom has

an occupied electronic state above the Fermi level of the metal, and thus full or partialelectronic charge transfer from the adatom to the metal will occur, causing a net positivecharge of the adatom, reducing the work function. Right: The adatom has an unoccupiedstate slightly below the Fermi level, which causes electron transfer from the metal to the

atom and an increase of the work function


the oxide layer surface and retain its positive charge, thereby affecting the

potential barriers such as to lower the work function. If the insulating layer is

thin, not only field emission, but also violent electrical ‘‘breakdown’’ of the

insulating layer may occur. ‘‘Electrical breakdown’’ refers to a phenomenon

where the previously insulating path becomes suddenly conducting, which is

usually associated with violent repositioning of atoms along the insulating path.

The energy for this repositioning of atoms is provided by the strong electric field.In a typical cathode situation, which will be discussed later in greater detail,

ions arrive from the plasma after they have transitioned the cathode sheath.

They have therefore sufficient kinetic energy to displace a few atoms of the oxide

or similar surface layer. The positive charge can cause a strong local rearrange-

ment of the electric charge distribution, the work function can be locally low-

ered, electrons emitted, and atoms desorbed. In fact, the presence of plasma in a

real cathode situation makes the situation much more complicated as described

so far because there are a number of processes contributing to the dynamics of

adsorption and desorption, including (but not limited to) desorption induced by

incident ions, electrons, photons, and energetic neutrals.Even the strong electric field by itself is able to affect the balance of adsorp-

tion and desorption, as illustrated in Figure 3.9. The formation of a potential

minimum resulted from attractive and repulsive forces (Figure 3.7). In the

presence of a strong external field, the resulting total potential is ‘‘bent’’ down

and thus the minimum is much shallower or not present all at, leading to field-

induced desorption or field evaporation.

3.3.3 The Role of Surface Roughness

Previous considerations assumed that the cathode surface was smooth. They did

not account for the effects of the periodicity of the lattice, atomic scale steps, and

Fig. 3.9. Potential illustrating field-induced desorption or field evaporation. In contrastto Figure 3.7, where the formation of a potential minimum resulted from attractive and

repulsive forces, the presence of a strong external field ‘‘bends’’ the total potential andthus the minimum is much shallower or not existent


larger-scale roughness. A decreasing atomic density reduces the dipole createdby electrons ‘‘spilling out’’ beyond the nominal surface [21], hence reducing thework function. Surface steps on the atomic scale also reduce the work function.These effects on the atomic scale reduce � typically by a few tenths of oneelectron-volt.

Real surfaces generally show much larger than atomic scale roughness. Evenin cases where the initial roughness is only on the atomic scale, the action ofcathode spots will cause roughening of the surface on the nanometer or evenmicron scale. This much larger-scale roughening leads to an enhancement of thelocal surface electric field strength and to non-uniform ion bombardment asillustrated by the cartoon (Figure 3.10). The influence of surface roughness onthe emission properties of metals was recognized early by Walter Schottky [25],who later became famous for his work on semiconductors.

The enhancement of the field strength of real surfaces is often captured by thead hoc introduction of a field enhancement factor, �, such that

Ereal ¼ �E0; (3:43)

where E0 is the electric field on the surface if the surface is atomically smooth andfree of non-metallic contamination. The value of � can be high and may exceed100 (see the review by Farall [26]). Experimental studies on field emission indi-cated that � appeared to be greater than 1,000 in some cases, which implied thateither the geometry of the electron-emitting centers is extremely pointed, needle-like, or that other factors play an important role. Although extremely acute shapesof emitters have been detected [27], their general presence appeared unlikely, andtherefore Latham [28] suggested that experimental data on � usually include notonly roughness effects but also the influence of non-metallic layers and dielectricparticles or inclusions in the surface layer as investigated earlier by Cox [29].

3.4 Theory of Collective Electron Emission Processes:

Non-stationary Models

3.4.1 Ion-Enhanced Thermo-field Emission

Until now, in describing the theory of electron emission due to high cathodetemperature, high electric surface field, or both, temperature and field were

Fig. 3.10. Illustration of non-uniform ion flux to a cathode (or other negatively biased

metal) and field enhancement. The non-uniformity is associated with a roughness that ismuch larger than atomic scale. Arrows indicate ion trajectories

3.4 Theory of Collective Electron Emission Processes: Non-stationary Models 95

assumed as given and stationary. For the case of a cathodic arc, the surface field

exists due to a space-charge layer or ‘‘sheath,’’ which is not stationary. Consider-

ing the nature of changes more closely one may recognize at least two timescales.One timescale is associated with changes of the sheath, which can be mainly

attributed to changes of the plasma density, and one may expect here changes

measured in nanoseconds. These are quasi-stationary changes on a timescale of

an actual electron emission event (femtoseconds).A second timescale and a strong modification of the original thermo-field

emission picture comes into play when considering the electric field of individual

ions. Ecker and Muller [30] proposed to modify the thermo-field mechanism for

the case of an arc by not just considering the average field of the sheath but the

actual, momentary field caused by an ion coming from the plasma and

approaching the surface. They showed that thermo-field emission can be sig-

nificantly enhanced without having to enhance the cathode temperature or

surface field.The deformation of the potential barrier by the approaching ion is shown in

Figure 3.11. The ion can only be effective in a narrow range of about 1nm of

distance from the surface. At large distances (4 5nm), the deformation of the

potential barrier by the ion is negligible, and at small distances (< 0.4nm), the ion

captures one or more electrons (if it was multiply charged) and becomes neutralized

(Figure 3.12). The timescale of action is thus t �s=vi 1 nm�104m=s ¼ 100 fs.

According to calculations for copper by Vasenin [31], ion enhancement of the

thermo-field mechanism can be significant, even exceeding a yield of 10 electrons

per incident ion, when the system is already in a highly emissive state, i.e., at

temperatures 4 4,000K and fields 4 109 V/m. Despite the individual nature of

the ion-field enhancement effect, ion-enhancement thermo-field emission belongs to

the group of collective emission mechanisms, and the discharge is an arc, not a glow

discharge, as explained in the Introduction to this chapter.

Fig. 3.11. Deformation of the potential barrier by an ion approaching the cathode surface(z = 0). Tunneling of electrons that can occur in the presence of a strong electric field isenhanced when the ion is sufficiently close to the surface, narrowing and lowering the

barrier. The ion charge is neutralized by capturing electron(s) and the enhancing effect isterminated


Coulombe and Meunier expanded such consideration to the case when

arcs are operated at relatively high gas pressure, that is, when bombardment

of the cathode with gas ions plays an important role. They showed that the

Richardson equation for thermionic emission is inadequate [19]: field-enhanced

thermionic emission (Richardson equation) underestimates electron emission

Fig. 3.12. Illustration of the time-dependent potential when an ion approaches a cathode,

calculated for a Cu2+ ion and an average field of 1.4� 109 V/m (adapted fromFigure 6 of[31]); x is the distance from the surface, r is the radial distance from the projected impactlocation. At large distances (x4 5 nm), the deformation of the potential barrier by the ionis negligible, and at small distances (x < 0.4 nm), the ion captured one electron


by at least 20% compared to thermo-field emission [17] enhanced by slowingmoving ions [19].

3.4.2 The Existence of a Critical Current Density

The emission of electrons by high temperature and high field will become intenseas the surface temperature and electric field strength Ereal approach typically2,000K and 109 V/m, respectively. However, none of the emission equations(3.7), (3.14), (3.16), or (3.21) indicate that there might be an upper limit ofemission current density.

In the 1950s, field emission projection tubes became a popular tool [32] tostudy the effects of surface features on the distribution of the field emissioncurrent. Walter P. Dyke and his colleagues [33, 34] applied the technique tostudy the onset of vacuum discharges. They found that the tips of field emissioncathodes exploded in less than 1 ms when the current density of electron emissionreached a critical value of about 1012 A/m2. They concluded that thermal effectsof the emission current itself were responsible for the explosive destruction of thefield emission tip and the onset of a vacuum arc.

The experiments of Dyke and co-workers indicated that stationary modesmay exist at relatively low current densities while current densities exceeding1011–1012 A/m2 imply explosive destruction of the emitting material. It was clearthat there is a need to consider the time-dependent energy balance of cathodesduring emission and ultimately consider the energy situation in a local and time-dependent manner. The development of a non-stationary emission model was ofcourse much more challenging than the development of stationary modelsbecause relevant non-stationary models require not only the explicit introduc-tion of time, but also the consideration of at least two, and better three, spatialdimensions, the temperature dependence of material parameters, and phasetransitions.

3.4.3 The Tendency to Non-uniform Emission: Cathode Spots

It is interesting to notice that the emission equations for thermionic and fieldemission assign to the emission area a very different role than to the governingparameters, temperature and electric field, respectively. While an increase inarea, at otherwise constant conditions, increases the emission current linearly,increase in temperature or field affects the emission current exponentially. Sur-face temperature and field have a vastly greater effect on emission and the systemas a whole than an increase of the emitting surface. Since a moderate increase intemperature or field on a very small area requires less energy than even a some-what smaller increase on a large area, stability considerations based on mini-mum energy dissipation will clearly favor the formation of small-area, hot spots.Different geometries, thermal conduction conditions, and materials may lead to


solutions of the energy balance that are stationary or non-stationary. Thesedifferent solutions manifest themselves as arc modes, which are later discussed inSection 3.6.

3.4.4 Energy Balance Consideration for Cathodes

Heat Conduction Equation

It is clear that the highest current densities occur most readily when both hightemperature and high electric field strength are present. An attempt to describethe situation could start by writing the general heat conduction equation for thecathode:

C�@T r; tð Þ@t

�r � rT r; tð Þð Þ ¼ j r; tð ÞE r; tð Þ; (3:44)

where C is the specific heat capacity (in J/kg K), � is the mass density (in kg/m3),� is the thermal conductivity (in J/s m K), T r; tð Þ is the temperature field (in K),j r; tð Þ is the current density distribution inside the cathode, and E r; tð Þ is theelectric field inside the cathode.

Joule Heat

The expression on the right-hand side of (3.44) is Joule heating. Joule heat iscaused by transfer of kinetic energy of free electrons in the metal to phonons(lattice vibrations). Free electrons in the metal are accelerated by the electricfield, E, causing the current density

j ¼ �E: (3:45)

This expression is Ohm’s law, where � is the electrical conductivity. The heatproduced in the volume of the cathode (in J/s m3) is

Sjoule ¼ jE ¼ �E2 ¼ j2

�: (3:46)

Apart from energy transport (heat conduction) and dissipation (Joule heat) inthe cathode volume, the local temperature of the cathode surface is determinedby energy fluxes associated with the ion, atom, and electron fluxes, as well aswith radiation:

q � � rTð Þn¼ qi þ qa þ qe þ qrad: (3:47)

The subscript ‘‘n’’ refers to the surface normal, which is directed away for thecathode. The energy fluxes through the surface are expressed in J/s m2. Theyinclude heating and cooling terms depending on the sign of flux considered.Relation (3.47) represents a boundary condition for (3.44). The energy fluxterms are discussed below.


Ion Bombardment Heating

Ions leaving the surface will remove energy from the surface, and ions arrivingwill bring energy to it. The dominance of one or the other depends on the stage ofdevelopment of the cathode spot. Ions arriving have acquired kinetic energy inthe space-charge layer, i.e., in the sheath, which is present between cathodesurface and quasi-neutral plasma. The voltage drop of the sheath is called thecathode fall, here designated as Vc. The energy density associated with arrivingions can be written as

qheati �

X

Q

j arriveQ

eQeQVc þ EQ �Q�þ Ecoh

� �; (3:48)

where the summation is over all charge states Q ¼ 1;2;3;:::;Qmax present in the

plasma. The term j arriveQ

.eQ is the particle current density of ions of charge state

Q. The second and third terms in the parenthesis are the total ionization energyof the ion minus the work function of the electron(s) needed to neutralize the ionwhen it arrives at the metal surface. The total ionization energy of a Q-chargedion can be calculated as the sum of all energies in stepwise ionization:

EQ ¼XQ�1

i¼0Ei (3:49)

where E0 is the energy needed to produce a singly charged ion from a neutral, E1

is the energy needed to produce a doubly charged ion from a singly charged, etc.The last term of (3.48) is the cohesive energy, which is defined as the energyneeded to remove an individual atom from its bonded position in the solid to anisolated position at infinite distance. The cohesive energy contributes to cathodeheating only when the ion condenses on the cathode and actually becomes partof it. Only a fraction 0 1 is actually accommodated; the rest, � 1,returns to the plasma after its charge is neutralized. For that reason, the accom-modation (or sticking) coefficient was introduced to (3.48).

The energetics of ions arriving at a surface is also important for film deposi-tion on a substrate, and therefore the situation is further considered in energeticcondensation of thin films (Chapter 8).

Ion Emission Cooling

Depending on the stage of spot development (see Section 3.4.8), ionsmaymainlyleave the cathode rather than arrive. This is especially true for stage (ii), theexplosive stage. In fact, the time-averaged net flux of ions is from, not to, thecathode surface, which is sometimes labeled as ‘‘anomalous’’ ion emission [35].In contrast to most other discharges, cathodic arcs are characterized by the netflux of positive ions moving away from the cathode, which is ultimately asso-ciated with the explosive nature of the cathode processes. The existence of spot


development stages is considered in the ecton model, and spatial and temporalsuperposition of spots leads to fractal features of cathode phenomena.

The cooling of the cathode by ‘‘anomalous’’ ion emission can be described by

qcooli � �X

Q

j emitQ

eQ2kTc þ EQ �Q�þ Ecoh

� �: (3:50)

This expression contains terms similar to (3.48); however, the kinetic energyterms are now associated with the temperature of the cathode, Tc. In writingdown this expression it is implied that the ions are formed in the explosiveprocess, hence their ionization energy is taken from the cathode.

This view on spot modeling with ion cooling is, however, not generallyaccepted since it is possible to consider that most ions are formed at somedistance from the cathode surface by ionization of an intense flux of atomvapor [36]. Under such conditions, ion emission cooling would not play a rolebut the energy removed by evaporation.

Atom Evaporation Cooling

For ions produced by ionizing the flow of evaporated atoms, one could omit theterms EQ �Q� in the parenthesis of (3.50), arriving at a corresponding expres-sion describing cooling caused by evaporation of atoms from the surface:

qcoola � �Jevap0 2kTc þ Ecohð Þ; (3:51)

where J evap0 is the flux density of evaporating atoms. In evaporation equilibrium,

the number of evaporating and condensing atoms are equal and the net flux ofatoms is zero:

na

ffiffiffiffiffiffiffiffiffiffiffikT

2pma

s

¼ pvaporffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pmakTp : (3:52)

In equilibrium, the temperature of atoms is equal to the temperature of theevaporating surface and T does not need to have an index. Following Lang-muir’s argument, the flux of evaporated atoms does not depend on whether ornot condensation is actually happening. Therefore, the evaporation rate canalways be related to the material’s equilibrium vapor pressure, and one maywrite

Jevap0 ¼ pvaporffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pmakTc

p : (3:53)

The vapor pressure increases approximately exponentially with surface tempera-ture. The material-dependent data curves were tabulated by Honig [37]. Oneshould note that the vapor pressure curves go smoothly through the meltingtemperature and therefore, from an evaporation point of view, the phase state ofthe cathode does not matter. However, this state matters a lot when we consider


the pressure of the plasma on the cathode: a liquid surface will yield, and themost apparent result is the generation of macroparticles which in turn becomesources of vapor. Macroparticles are further described in Chapter 6.

In the framework of a fractal model, evaporation cooling can be neglectedduring the explosive spot stage since the emitted material is fully ionized andthus already taken into account in the ion flux term. In other spot stages or,equivalently, outside the spot center, evaporation may occur but its energeticconsequences are small compared to other forms of energy flux due to the verystrong temperature dependence of the vapor pressure.

Atom Condensation Heating

Analogous to (3.51), the energy brought to the cathode by condensing neutralatoms is

qheata � Jcond0 2kTa þ Ecohð Þ; (3:54)

where Ta is the atom or vapor temperature. In the explosive stage, no atoms canflow against the stream of material due to collisions. Atoms may condense inother spot stages (or equivalently, outside the spot center) but the associatedenergy flux can be neglected compared to other energy fluxes. For example,returning ions will have gained energy by acceleration in the field of the cathodesheath but (neutral) atoms do not ‘‘see’’ this field.

Electron Emission Cooling

The emission and return of electrons can have an important influence on theenergy balance of the cathode. As extensively discussed before (Section 3.3.1),electrons are confined inside themetal by a potential barrier, whose height abovethe Fermi level is �, the work function, or better �S, the Schottky-corrected workfunction, see (3.13). For a hot cathode, electrons can leave the metal classically,going over the barrier and hence carrying away the energy:

qcoole � � jemite

e2kTc þ �Sð Þ: (3:55)

If the emitting cathode location is still cold (in terms of thermionic or thermo-field emission), significant emission can only occur via field emission in a strongelectric field. As discussed before, electrons tunnel quantum-mechanicallythrough the barrier, which has become narrow by its deformation in the strongfield. Now something strange can happen, namely that the emission of elec-trons can actually lead to heating rather than cooling, which is known as theNottingham effect [13, 38]. This becomes clear if one recalls that electrons in ametal have a Fermi distribution, as was illustrated in Figure 3.2. Electronsbelow the Fermi level may tunnel and leave the metal; their replacementfrom the current supply fills the electron ‘‘sea’’ at the Fermi level. The energy


difference between the electron lost and the electron replaced constitutes asmall heat gain. As the cathode heats up, this heating becomes less and even-tually reverses to the more familiar, classical cooling. The inversion tempera-ture, T �, at which heating turns into cooling has been determined to [13, 39]

T � ¼ 5:67� 10�7E

�1=2t yð Þ; (3:56)

where T � is in K, E is in V/m, � is in eV, and t yð Þ is the elliptical functionintroduced before and tabulated in Table 3.1. From this expression one can seethat Nottingham heating can only be important in the pre-explosion stage, whenthe local cathode surface temperature is still low. Therefore, one needs toconsider the Nottingham effect only for the onset of thermal runaway and theearly development of a local emission center.

Heating by Returning Electrons

While the net electron current is away from the cathode and electrons areaccelerated in the cathode fall, electron–electron interaction in the dense plasmawill quickly lead to a Maxwellian energy distribution. Electrons in the energetictail of the distribution may have enough energy to return to the cathode,essentially delivering the work function energy

qheate � j returne

e�S: (3:57)

The return current,

j returne � ene

ffiffiffiffiffiffiffiffiffiffiffikTe

2pme

s

exp � eVc

kTe

� �; (3:58)

contains an extremely small exponential factor because �eVc=kTeð Þ �10.Therefore, heating by return electrons can be neglected. In models that assumevapor ionization in a more or less pronounced potential hump, the electronreturn current is even smaller.

Radiation Cooling

The cathode spot is bright and obviously energy is removed from the cathode viaradiation. In a rough approximation, one may assume that the spot area is ablack body radiator whose power density is given by

qcoolrad ¼ "c�SBT4c ; (3:59)

where "c is the surface emissivity ("c ¼ 1 if the surface was a true black body, but

realistically 05"c51), �SB¼ 5:67� 10�8W=m2 K4 is the Stefan–Boltzmann

constant, and Tc is the temperature of the emitting cathode surface. Here and


elsewhere we have a conceptual problem since Tc is not well defined for the caseof a microexplosion. For the sake of an estimate, one may assume that Tc

exceeds the melting temperature. Even for refractorymetals and given the strong

T 4c dependence, one can easily determine that radiation cooling is small com-

pared to other energy terms.

Radiation Heating from Plasma

The light seen from a cathode spot is mainly emitted from the spot plasma. Thestatement above in the first paragraph of ‘‘radiation cooling’’ was therefore a bitdeceptive! The plasma formed in a microexplosion is very dense and opticallythick, hiding (shielding) the radiation emitted from the cathode-spot surface.The term optically thick refers to a medium in which the mean free path ofphotons is small compared to the physical size of the medium. Photons areabsorbed and re-emitted again and again, and photon transport resemblesdiffusion [40]. With its expansion, however, the plasma becomes quickly trans-parent or optically thin.

Radiation coming from the plasma will be in part reflected and in partabsorbed; only the latter contributes to heating of the cathode. A quantitativedetermination of this heating is difficult because it involves the black bodyradiation of the transient plasma of the explosive stage of the cathode spotand line radiation from the expanding plasma corresponding to later stages.Only the black body radiation with temperature of a few eV (say,Tc � 50; 000K)would be noticeable; however, even this can be neglected in the overall balance.

3.4.5 Stages of an Emission Center

The previous discussion already indicated that electron emission becomes non-stationary and localized. The terms ‘‘spot’’ and ‘‘emission center’’ were used,calling for more refined considerations, which will be given later in the frame-work of a fractal approach. At this point one may think of a spot or an emissioncenter as a location on the cathode surface where one can describe the evolutionof electron and plasma generation. The evolution may be divided into fourstages: (i) the pre-explosion stage, (ii) the explosive emission stage, (iii) theimmediate post-explosion stage, where cooldown has started but electron emis-sion and evaporation are still large, and (iv) the final cooldown stage. Each of thefour stages is highly dynamic. In the following paragraphs, these stages arebriefly described, followed by a discussion of plasma and sheath properties.

In the beginning of the pre-explosion stage, the cathode surface has assumedsurface conditions determined by its history, such as mechanical and heattreatment, and the exposure to specific intentional or residual gas conditions.Since we consider the operation of an arc (and not the specifics of the initial arctriggering), plasma is already generated at some distance, causing cathode loca-tions to be exposed to bombardment by ions, accompanied by a flux of electrons,


atoms, and radiation. Furthermore, the temperature of the solid may be increas-

ing, e.g., due to heat conduction and ion bombardment.Each location on the surface might be a candidate for the ignition of an

emission center but in reality each location has its own history and properties

such as the work function. For example, the conditions are different at grain

boundaries and in the middle of a grain. Oxides and other dielectrics might be

present, and surface roughness will certainly give each location unique

properties.Suppose one could pick a location that is going to experience explosive

emission. The conditions will be such that the local energy input will be higher

than at neighboring locations due to its specific properties and its relation to the

plasma and sheath conditions produced by predecessor emission sites. The

conditions are indeed favorable if the local work function is low and the field

is enhanced due to the presence of dielectric contamination and/or the presence

of micro- or nanoprotrusions. If such favorable conditions are coupled to a very

high electric field strength (e.g., thin sheath due to high plasma density) and a

high intensity of ion bombardment, the local energy input can lead to electron

emission with thermal runaway, bringing the location to stage (ii), characterized

by explosive electron emission. This stage is at the heart of the ‘‘ecton’’ model

developed by Mesyats and co-workers [41, 42]. Thermal runaway and ecton

model are discussed later in this chapter. The microexplosion causes destruction

(erosion) of a microvolume, which is later evident as a crater on the cathode

surface. Theoretical models of cathode-spot development differ in the literature,

although most agree that repetitive ignition of microexplosions is real and well

supported by experimental evidence. The main discontent is about the duration

and relative importance of explosion and post-explosion stages.In one view, each microexplosion is immediately followed by the next micro-

explosion, and therefore the cathode operation is based on a rapid sequence of

microexplosions, each being on a timescale of the order of 10–8 s [41, 42]. In this

view, stage (iii) is of relatively little importance.High-resolution, fast optical diagnostics support this view, at least the notion

that the sequence of explosive events is indeed rapid. For example, very fast

optical imaging of very low current arcs, 3–12 A, only shows bursts of light every

50–70 ns with the most intense phases having a duration of 10–20 ns [43].Another view considers the explosive stage (ii) as a short, transient, beginning

stage for a much longer, quasi-steady-state, post-explosion stage in which elec-

trons are emitted from the hot, liquid metal layer of the freshly created crater

formed under the action of dense plasma. In this view, cathode material is

evaporated and becomes ionized very close to the surface due to the intense

electron beam formed in the thin cathode sheath. Electrons in this beam have the

energy corresponding to the cathode fall (about 20V) and their current density is

determined by field-enhanced thermionic emission. Most ions are formed by

electron–atom interaction in an electron beam relaxation zone in close proximity

to the cathode surface [36].


Putting these differences aside, for the time being, it is clear that a fourth, finalstage must exist in which electron emission and evaporation have ceased becausethe thermal conduction has led to an increase of spot area, lowered the powerdensity, and hence lowered the surface temperature. The explosively formedplasma has expanded, its density is lowered, therefore the cathode sheath thick-ness has increased, and the surface field is reduced. Despite lower cathodesurface temperature and lower field strength, this stage may be important tothe overall cathode erosion since the hot surface may still deliver metal vapor,especially when the cathode material is of high vapor pressure [44].

3.4.6 Plasma Jets, Sheaths, and Their Relevance to Spot Ignition and Stages

of Development

Neglecting the small voltage drop inside the cathode, one may state that thecathode is at the cathode potential and the plasma far from the surface is atplasma potential. ‘‘Far’’ can be understood as a distance much larger than thespot size and larger than the greatest cathode sheath thickness. The potentialdifference

�V ¼ Vcath � Vpl (3:60)

is located very close to the cathode surface and is generally known as the cathodefall. The thickness of the sheath, across which the potential falls, dependsstrongly on the local plasma density. Local emission and explosive processesimmediately imply that we deal with a time-dependent and non-uniform dis-tribution of plasma density and sheath thickness. The situation is quite differentthan often simplifyingly illustrated: the sheath edge or boundary is not at aconstant distance from the surface. Quite contrarily, the sheath boundarydepends on the local plasma conditions and changes rapidly with the evolutionof the emission center.

Let us consider an instantaneous snapshot of the near-cathode zone andpreliminarily adopt the concept of the Child sheath (Appendix A):

sChild ¼ffiffiffi2p

3lDe

2e �Vj jkTe

� �3=4

: (3:61)

The Child sheath thickness scales with the Debye length lDe ¼ "0kTe

�nee

2� �1=2

,which varies greatly over the surface. The Child sheath thickness depends onthe voltage drop, �V, and on the local electron density and, to a lesser degree,electron temperature. One can therefore immediately see that different locationshave different sheath thickness: a location of denser plasmas has a thinner sheaththan the surrounding locations.

Although these considerations appear intuitively right, they do not stand arigorous proof because the conditions of validity for (3.61) do not apply for alllocations, or equivalently, stages of spot development. One of the assumptionsin the derivation of (3.61) was that ions move from the sheath boundary through


the sheath toward the cathode surface. However, the net flux of a cathodic arc

process points in the opposite direction. The assumption of arriving ions is

greatly violated at least in stage (ii), the explosive stage, where the phase transi-

tions solid!liquid!gas!plasma occur rapidly in an expanding volume. The

voltage difference �V drops mainly in the dense, non-ideal plasma because the

conductivity of the non-ideal plasma is less than the conductivity of the metal

and the conductivity of the expanded, ideal plasma (‘‘valley of low conductivity’’

[45]; more about non-ideal plasmas are said in the next section).The plasma conditions change rapidly and therefore the sheath boundary

should be understood as a highly dynamic object with transient ‘‘holes.’’ A

‘‘hole’’ means that a location may exist without a sheath: the voltage is dropping

in non-ideal, quasi-neutral plasma, as opposed to a space-charge layer. The

sheath holes are at locations where dense plasma of microexplosions can be

found, hence they exist only at some locations for a very short time (nanose-

conds). Figure 3.13 shows a cartoon snapshot of such situation. This description

illustrates the difficulties of modeling cathode processes of cathodic arcs: the

different stages of spots and fragments require different model approaches and

different scales. Explosive models need to be matched with models describing

simultaneously occurring processes far from the explosive center. These differ-

ent processes are coupled; they operate electrically in parallel and create bound-

ary conditions for each other.At this point some more remarks should be made about model assumptions.

In order to make the difficult situations tractable, simplifying assumptions are

made on the structure of layers and geometry of emission sites. Furthermore,

sometimes, models are not based on first principle equations but on solutions of

fundamental equations that apply to certain conditions.For example, instead of solving the Poisson equation, a second-order differ-

ential equation, special solutions such as the Child–Langmuir law for the current

and the Mackeown equation for the electric surface field are often utilized.

Mackeown’s equation [46, 47] in one dimension is simply

Fig. 3.13. Schematic of ‘‘holes’’ in the cathode sheath: these are locations where denseplasma of microexplosions is found; ‘‘sheath holes’’ exist only for a very short time

(nanoseconds)


E surface ¼@Vc

@z

��surface

� 4

3

Vc

sChild; (3:62)

where Vc � 20V is the cathode fall and sChild is the Child sheath thickness,

(3.61). However, these special solutions are only valid under certain conditions,

such as the net ion flux is going toward the surface, as already mentioned. Such

conditions may be satisfied far from the spot or, equivalently, at times other than

the explosive stage.Simplifying assumptions are also made in models focusing on the explosive

stage. Most importantly, to model thermal runaway, the field enhancement

needs to be known, and therefore assumptions are made about the shape of a

‘‘typical’’ emission site. Very popular is the cone shape because a cone has a very

high field enhancement factor at its tip, and furthermore it is argued that cones

are naturally formed when a liquid surface is subject to strong electric fields

(Taylor cones) and dynamic pressure (nonlinear surface waves) [48].The sheath thickness is of critical importance to the ignition of an emission

center because it determines the surface electric field, which needs to be suffi-

ciently high to cause thermal runaway at this location. Even as the Child solution

(3.61) and the Mackeown field (3.62) are not applicable, one can qualitatively

say that high plasma density is associated with thinner sheath and higher surface

field strength. Therefore, as we look for reasons why a potential emission site

actually becomes an emission site, we need to consider the site’s surface conditions

as well as the evolution of plasma above the surface. This approach will naturally

lead to understanding of random versus ‘‘steered’’ motion of cathode spots.In the previous section on stages of development, it was already mentioned that

ignition of a new emission site occurs when a location with favorable surface

conditions is exposed to dense plasma. The dense plasma simultaneously produces

two important effects: one is the shrinking of the sheath thickness and the asso-

ciated increase in electric surface field and the other is an increase in ion bom-

bardment heating. The combination of both gives rise to intensified local energy

input, which leads to a microexplosion if the energy input rate exceeds the energy

removal rate, as will be discussed in the framework of explosive electron emission

(see (3.63)). At this point it should only be mentioned that experiments by

Puchkarev and Bochkarev [49] and simulation byUimanov [50] provided evidence

that ion bombardment heating is critical for the formation of the explosive emis-

sion stage. Active emission sites emit plasma that is rather non-uniform on the

microscopic scale (jets) due to focusing in the strongmagnetic field associated with

the arc’s high current density. These micro-plasma jets provide the ion bombard-

ment and field enhancement conditions for potential emission sites. In the absence

of an external magnetic field, the self-field is rather symmetric, and there is no

preferred direction for the emission of microjets. Hence the ignition of new emis-

sion sites is equally likely in all directions from the active site. If an external

magnetic field is applied transverse to the surface normal, the symmetry is broken,

and one should expect a preferred direction of plasma jets and spot ignition.


3.4.7 Explosive Electron Emission and Ecton Model

The central stage in the development of emission centers is stage (ii), the

explosive stage. The idea of explosive electron emission was developed by

Mesyats and co-workers including Bugaev, Litvinov, and Proskurovsky

[51–53]. The term ‘‘explosive electron emission,’’ sometimes with the acronym

‘‘EEE,’’ may refer to the whole development cycle of an explosive center, or

specifically to the explosive stage.Already in his early work in the 1960s, Mesyats recognized the similarity

between metal!plasma phase transitions of exploding wires and cathodic arcs.

Based on much improved experimental data, Mesyats introduced a rather

specific model of explosive electron emission in the 1990s, the ‘‘ecton model’’

[54, 55], which is based on the explosion of a liquid metal cone formed by the

action of a strong electric field on the hot cathode surface.The term ‘‘ecton’’ refers to an ‘‘explosion center’’ with the ending ‘‘ton’’ in

analogy to other particles or quasi-particles (like photon, proton, exciton, etc.).

In doing so, Mesyats wanted to emphasize the discrete, ‘‘quantum-like’’ appear-

ance of each explosive event [35, 41, 42, 56]. Interestingly, one can also see the

collective character of electron emission (recall: Hantzsche defined an arc cath-

ode mechanism to be a collective phenomenon, see the discussion at the begin-

ning of this chapter). According to the ecton model, each ecton liberates about

1011 electrons in an explosion with duration of the order of 10 ns. The micro-

explosion also produces plasma of the cathode material and generates the

conditions for the ignition of the next ecton.Mesyats refers in his work generally

to emission centers of cathode spots; the physics applies to individual emission

centers or cells or spot fragments in Kesaev’s [57] and Juttner’s [58]

classifications.The explosive emission concept will now be described in somewhat greater

depth because it contains processes that explain a number of features of the

cathodic arc plasma. These features (supersonic ion velocities, multiple charge

states, etc.) distinguish cathodic arc plasmas from most other plasmas; and they

are the reason that cathodic arc plasma deposition is an energetic condensation

process leading to films that are denser than most other processes.A microscopically small volume, such as a microprotrusion or particular

volume under an oxide layer, will explode if the rate of specific energy input,

dw=dt, is much greater than the maximum rate of heat removal. The latter can be

expressed as the energy of sublimation or cohesive energy, Ec, divided by the

characteristic time of energy removal, t, [54], hence the condition for explosion is

dw

dt� Ec

t: (3:63)

The cohesive energy can be expressed as energy per mass (J/kg) or per particle

(eV/particle), and the characteristic time of energy removal can be determined by

t ¼ d=vs; (3:64)


where d is the characteristic (linear) size of the microvolume and vs is the speed of

sound in the cathode. The conditions so far are rather general and not necessa-

rily bound to the conical shape of an emitter. Explosive emission may occur on

rather plane surfaces [49] as long as condition (3.63) is meaningful and satisfied.

If heat removal is exclusively by heat conduction, the linear dimension of the

exploding microvolume is

d5ffiffiffiffiffiatp

; (3:65)

where a is the thermal diffusivity. For example, withwe 107 J=kg, vs 103 m=s,

d 10�6 m, one obtains dw=dt� 1016 J=kg s.Explosive phase transitions are known from wire explosions [59]. A segment

of wire can be considered equivalent to a current-carrying microprotrusion on

the cathode surface. A wire (or microprotrusion) will explode with a delay time,

td, if a thermal runaway instability occurs [60]. Joule heating is proportional to

the current density and voltage drop; the latter, in turn, is proportional to the

current density and resistance of the wire segment (or microprotrusion). For

metals, the resistance increases with temperature and, provided the power source

can deliver greater power at increased voltage, the Joule energy dissipation

increases with increasing temperature. This, in turn, increases the temperature,

which increases the energy dissipation, etc., and thus the runaway feedback

loop closes until the wire segment (or microprotrusion) is destroyed by a

microexplosion.From the theory of wire explosions [61], the current density, j, and explosion

delay time, td, satisfy

ðtd

0

j2dt ¼ h; (3:66)

where h is called the specific action whose value depends on the cathode material

but is approximately independent of current density, wire cross-section, or other

discharge quantities (Table 3.4).

Table 3.4. Specific action, h, forselected materials. (From [41, 62])

Material h (A2s/m4)

C 1.8Al 18Fe 14

Ni 19Cu 41Ag 28Au 18


According to the ecton model [41, 54, 56], thermal runaway occurs on

microprotrusions until they explode with a delay time associated with the

material-specific action. Cathodic arc operation includes plasma production

and electron emission via a rapid sequence of microexplosions. This concept

may appear questionable because it explicitly calls for explosions of micropro-

trusions. Different materials have certainly different densities of such protru-

sions, and of course they are destroyed by the explosive processes. Therefore, to

be consistent, the ecton model must include not only the explosion but the

formation of microprotrusions or similar explosion-promoting structures or

other conditions that can lead to thermal runaway in a limited volume. The

latter may occur on any metal surface if plasma microjets provide intense local

ion bombardment and high electric surface field [49].Not all modeling work is based on explosive emission theory (used widely in

this book), where electron current actually exceeds the arc current because it

needs to compensate for ions going the ‘‘wrong’’ way (from cathode to anode). In

a more traditional one-dimensional model, the current transfer at the cathode

surface is composed of electron emission current and ion return current. Using

this and other assumptions, Beilis [63] calculated that multiply charged ions may

be formed for refractory metals at a relatively small electron-to-ion current ratio

of 0.7–0.9. A similar model was developed by Coulombe and Meunier [64] for

the operation of a copper arc with current densities in the range from 108 A/m2

(upper limit for non-vaporizing cathode models) to 4� 1010 A/m2. Their results

showed that current densities greater than 1010 A/m2 can only be accounted for

with metal plasma pressures exceeding 35 atm and electron temperatures ran-

ging from 1 to 2 eV. The current transfer to the cathode is mainly assumed by the

ions at relatively low current densities (<1010 A/m2) and by the thermo-field

electrons for higher current densities. The heat flux to the cathode surface under

the spots is mainly due to the flux of returning ions and ranges from about 1010

to 1011 W/m2 for current densities ranging from about 109 to 1010 A/m2.

3.4.8 Explosive Electron Emission on a Cathode with Metallic Surfaces

For fragments of spot type 2, i.e., emission sites on clean metal cathode surfaces

as further discussed below, the formation of microprotrusions is not difficult to

recognize when investigating the crater traces left by the microexplosions (right

side of Figure 3.14). Themicroexplosion produces a thin layer of molten cathode

material, which yields to the high plasma pressure. The liquid material is ejected

from the explosion crater and is rapidly quenched, producing macroparticles

(see Chapter 6) and microprotrusions, which can serve as new ignition points.

One of them will be most suited to go through thermal runaway, leading to the

next microexplosion. According to this picture, the location of the next explo-

sion is displaced by about one crater radius from the location of the predecessor

[65]. It is common for this arc spot type that long chains of craters are formed


(Figure 3.14, right). These chains of craters show a preferred (retrograde) direc-

tion when an external magnetic field is present.At first sight, the situation is less obvious for atomically smooth cathodes or

liquid cathode surfaces like with mercury or gallium or cathodes with non-

metallic layers (type 1 cathode spots). Here, additional processes need to be

considered.Let us consider a liquidmetal surface subject to a strong electric field. Already

Tonks considered in 1930 that the electrostatic force on a liquid metal surface

may overcome the stabilizing forces of surface tension and gravitation [66].

A surface instability can lead to the development of liquid metal cones,5 which

could well serve as the microprotrusions needed to maintain the explosive

electron emission process. Figure 3.15 shows microprotrusions formed on a

liquid metal surface in a strong electric field [48]. Although this figure was

obtained by intense and quenched electron bombardment in a specially con-

structed electron microscope, similar conditions may occur on the surface in the

vicinity of an emission center because ions from the dense plasma bombard the

surface, corresponding to stage (i) of the fragment development.Numerical modeling needs to include the temporal evolution (time is an

explicit variable) and, in order to keep the problem manageable, simplifications

Fig. 3.14. Track of erosion craters left by spots of type 1 (left) and type 2 (right); a

transverse magnetic field was applied to ‘‘drive’’ apparent spot motion leaving a ratherstraight trace. (Photos courtesy of B. Juttner)

5 Very sharp liquid metal cones (Taylor cones) are used in liquid metal ion sources

(LMIS) [67, 68] where field evaporation and ionization occurs for use in focus ionbeams (FIBs), for example. However, the polarity of those devices is opposite to ourcathodic arc configuration, and the currents are in the range of hundreds of nA per tip,i.e., many orders of magnitude less than arc currents.


are done in terms of geometry, for example by assuming a slender cone with givenmaterial properties. The explosive stage was shown to be consistent with a timeregime of 1–10ns, peak current density of up to 1013 A/m2, and surface field at thetip of the cone of up to 1010 V/m [69], although there is evidence for even higher,transient current densities in the sub-nanosecond time regime and much lowervalues when looking at ‘‘most probable’’ quantities [70]; this wandering overseveral orders of magnitude is re-considered in the fractal model, Section 3.5.

Numerical simulation using a time-dependent, two-dimensional hydrody-namic model indicated that the current distribution at the end of the explosivestage is ring-link, rather than peaked in the center of the emission site [71, 72]. Inthe absence of a transverse magnetic field, any place on the rim of the newlyformed crater could turn into a new, active emission site, whereas the current-carrying ring is not symmetric when the plasma above the emission site is bent bythe transverse magnetic field. This asymmetry can contribute to the non-iso-tropic probability for igniting new sites of type 2 spots.

3.4.9 Explosive Electron Emission on a Cathode with Non-metallic Surfaces

Turning to surfaces with non-metallic (e.g., oxide) layers, one should recall thata microexplosion occurs when the rate of specific energy input, dw=dt, is muchgreater than the maximum rate of heat removal. This condition does notnecessarily imply a conical shape of the emitting surface. More importantly,the rate of rise of local temperature must exceed a threshold value. By limitingthe explosion time to less than 100 ns (a reasonable though arbitrary value),Abbaoui and co-workers [73] calculated for a range of materials that the powerdensity must exceed about 5 � 1012 W/m2 otherwise the phase transitions will

Fig. 3.15. Microprotrusions formed on a liquid metal surface in a strong electric field.This photograph was obtained in a scanning electron microscope where a plane coppersurface was placed in a field of 107 V/m and exposed to plasma flow, which provided

intense heating that led to surface melting. When the flow was terminated, the surfaceprofile quenched, allowing us to observe quenched microprotrusions. (From [48])


not occur. Simulating local power input with a focused, 20-ns pulsed laser beam,Vogel andHoft [74] found that about 2� 1011W/m2 are need to obtain localizedmelting; however, in order to produce craters that look like arc craters one needsa power density in the range 1012–1013 W/m2.

As discussed in previous sections, the presence of non-metallic layers pro-foundly changes the electronic structure of the surface. The work functions canbe lowered, which is important because electron emission depends on the workfunction in a highly nonlinear manner. If there is insulating material on thesurface, it will charge up under ion bombardment and create strong field withinthe insulator, which in turn can lead to the enhancement of electron emissionand thermal runaway at some locations. (Similar processes can also occur onnegatively biased substrates leading to ‘‘arcing,’’ and the topic is therefore pickedup again at the beginning of Chapter 9.)

The rate of surface charging, field enhancement, and emission onset can bevery fast, which can be more important than the absolute energy input, as wasexpressed in (3.63). It is conceivable that several emission centers may switch onnearly simultaneously. Such multiple centers are electrically in parallel, henceshare the total arc current and the same total voltage drop. They ‘‘compete’’ forcurrent and energy dissipation. In the average, emission centers in this situationwill have less current per center, in agreement with observations. Emissioncenters that have high impedance will have less current and less heat dissipationand in absolute terms may not maintain the rate of rise required to fully formexplosion craters: such centers will stop producing phase transition of cathodematerial, they will ‘‘die’’ in an early stage. Indeed, electron microscope picturesof arc traces on ‘‘contaminated’’ (i.e., non-metallic) surfaces show craters of type1 spots dispersed over the cathode surface, with a large number of small and verysmall craters (diameter 1 mm and less, Figure 3.14 left). In this picture, the‘‘death’’ of an emission site is associated with the ignition, presence, and compe-tition of other emission sites. This is consistent with the observation that spots oftype 1 have faster apparent motion, need less voltage to ignite, erode less cathodematerial, emit less light, and produce a plasma that contains metal and non-metal species.

3.5 Fractal Spot Model

3.5.1 Introduction to Fractals

Cathodic arcs showmany features that suggest to model cathodic spot phenom-ena using the well-known theory of fractals [75, 76]. For example, random walkof cathode spots was discussed in the 1980s [77, 78], 1/f-noise of ion current hasbeen found in the late 1980s [79], arc traces by the spot’s random walk showed afractal dimensions of about 2 [80], and self-similarity in the patterns of emittedlight was recognized [81, 82]. As has been argued in a recent publication [3],fractal features are not superficial but fundamental to the nature of cathode


spots. After reviewing all evidence, one can agree with Mandelbrot’s sentimentquoted at the beginning of this chapter.

With this motivation, let us start with a brief introduction to the world ofpower laws, random colored noise, and fractals.

Many laws in physics are linear and periodic and show invariance to additivetranslation. However, not all physical phenomena can be described in thismanner; in fact, a great number of phenomena are nonlinear, aperiodic, andchaotic. In the 1980s, a branch of mathematics and physics started to flourish:the science of deterministic chaos and self-similar structures, dubbed ‘‘fractals’’by Benoit Mandelbrot [75]. Fractals are mathematical or physical objects invar-iant to scaling, which makes them ‘‘self-similar’’ to multiplicative changes ofscale. A self-similar object appears (approximately) unchanged after increasingor decreasing the scale of measurement and observation. Self-similarity may bediscrete or continuous, deterministic or probabilistic.

Power laws are an abundant source of self-similarity [76]. Consider thehomogenous power law

f xð Þ ¼ cx (3:67)

where c and are constants. It is self-similar because rescaling (i.e., multiplica-tion with a constant) preserves that f xð Þ is proportional to x albeit with adifferent constant of proportionality. A fruitful approach to fractal modelingis to look for power laws describing the physical phenomena.

Self-similarity can be mathematically exact and infinite or only approxi-mate and asymptotical. The latter applies to the physical world, and scalingcutoffs exist at the small scale and large scale. Mathematical objects are strictlyself-similar with infinite scaling; they are often named after their ‘‘inventors’’and have sometime colorful names, like Cantor sets, Julia sets, Koch flakes,Sierpinski gaskets, Mandelbrot trees, Farey trees, Arnold tongues, Devil’s stair-case [75, 76, 83].

Fractals can be characterized by dimensional measures, such as the Hausdorffdimension (named after Felix Hausdorff, 1868–1942). The fractal dimension isoften non-integer and smaller than the embedding topological dimension. Toillustrate the concept, Mandelbrot [75] used the now-classical question ‘‘Howlong is the coast of Great Britain?’’ The essence of this consideration is repro-duced here, having in mind that the question of current density and otherparameters should be considered in the same way. Mandelbrot showed that(i) the answer depends on the scale length of measurement and (ii) differentcoastlines have different values for the fractal dimensions, depending on their‘‘ruggedness.’’ More detail is revealed by ‘‘zooming in,’’ therefore, the finer thescale unit r, the longer the apparent coastline length L. If the coastline is self-similar, a power low can be found connecting the measured length L rð Þ with themeasuring scale unit r:

L rð Þ / r"; (3:68)

3.5 Fractal Spot Model 115

where "50 if L increases as r decreases. In contrast, if the coastline was smooth,the measured length should approach an asymptotic value when a smaller scaleis used, L is finite as r! 0, hence " ¼ 0. The Hausdorff dimension

D ¼ limr!0

logN rð Þlog 1=rð Þ (3:69)

can be introduced, where N rð Þ is the minimum number of circular areas ofdiameter r needed to cover the coastline. In this example,N rð Þ ¼ L rð Þ=r, leadingto

D ¼ 1� ": (3:70)

Because "50, D will exceed 1, which can be intuitively interpreted as a dimen-sion because a coastline is more than a one-dimensional object (straight line) andless than a two-dimensional object (filled area). Coastlines with only few baysand points have D just little over 1, while rugged coasts such as the coast ofBritain or Norway exceed 1 significantly.

Fourier transform of stochastic, fluctuating data often reveals that the Four-ier power spectrum follows a power law

F fð Þ 1=f ; (3:71)

where f stands for frequency. According to convention, the exponent deter-mines the color of random colored noise (RCN, [84]). The noise is white for ¼ 0 (i.e., it does not depend on frequency); ¼ 1 describes pink noise, which isoften found in physical systems of self-organized, barely stable structures [85]; ¼ 2 is brown (hinting at a relation to Brownian motion); and 42 refers toblack noise observed in self-similar systems that tend to have positive feedback,like the stock market.

3.5.2 Spatial Self-Similarity

The connection between discharge phenomena and fractals is well establishedfor branching and treeing of the conducting plasma channels in lightning andLichtenberg figures (Figure 3.16). Crystal growth by diffusion-limited aggrega-tion (DLA) leads to structures that look very similar to Lichtenberg figures aswell as spot traces obtained by high-current pulsed arcs. Therefore, it appears tobe instructional to consider well-established modeling of DLA.

In DLA, a single molecule performs a random walk until it bonds to anaggregate; it gets ‘‘stuck’’ and provides an attachment site for the next molecule.A kind of dendritic growth is observed because the molecule has a greaterprobability to attach near one of the tips of the fractal cluster than in the ‘‘fjords’’[76, 87]. Different sites have different probabilities for attachment, and theprobability for attachment decreases with increasing depth inside a ‘‘fjord.’’Figure 3.17 shows a simulation of a DLA growth cluster (left) and the pointsof enhanced attachment probability (right) [87]. Lichtenberg figures and


multi-spot arc traces have similar shapes, which is not accidental because all ofthese phenomena are governed by the Laplace equation for a potential. In DLA,the cluster is an equipotential surface and the gradient of the potential corre-sponds to the diffusion field. In discharge physics, the tips of a growing Lichten-berg figure and ignition locations of a developing multi-spot arc have the samepotential due to the high conductivity of the plasma channels.

Fig. 3.16. Lichtenberg figure: a pulse discharge on the surface of an insulator [86]

Fig. 3.17. Simulation of a DLA growth cluster (left) and the points of enhanced attach-ment probability (right) [87]


It should be noted that a multiplicative random process on a fractal, ratherthan on an interval, leads to amultifractal spectrum, a concept nowwidely used inturbulence research. Diffusion-limited aggregation is one example of a multi-fractal [76], and one should expect to discovermultifractal properties for arc spots.

One of the early works on spots and fractals is by Siemroth, Schulke, andWitke who recognized self-similarity in at least four levels of spatial resolution[81, 82]. According to their work, one might distinguish group-spot level, macro-spot level, microspot level, and crater level and find self-similar features in eachof those levels. In this chapter, and throughout the book, it is argued that thefractal cathode-spot model is fundamental both for spatial and temporal phe-nomena, while elementary processes like explosive events (ectons) represent thelower cutoff for self-similar description in time and space.

Stochastic self-similarity exists not just because we can ‘‘zoom’’ with differentresolution and find similar spot appearance; rather, the stochastic but field-dependent (deterministic) nature of spot ignition provides a physical basis for afractal model of cathodic arc spots. By incorporating the ‘‘quantum’’ of explo-sive emission (ectons) as the cutoff limit and considering stages of spot develop-ment, the ecton model and vapor ionization model can be reconciled withobservations on both short and long scales of time and space.

It should be noted that the non-zero, finite lifetime of excited states presents afundamental limit to ‘‘zoom in’’ with optical emission methods. The location ofde-excitation processes, in which observable photons are emitted, is shifted withrespect to the location of excitation, and therefore there is a fundamental limit ofresolution using emission methods [88–90]. Images of emission centers show adiameter of typically 100 mm or larger, even when the optical resolution is a fewmicrometers, while the diameter of the underlying erosion crater may be just afew micrometers or less. Of course, this has implication for the limits of currentdensity measurements [88–90]. Going beyond that, the non-zero lifetime ofexcited states represents a lower observation limit (cutoff) for the fractal cathodespot when observed by optical emission methods.

Self-similarity can also be found in traces left by cathodic arcs. This includesmacroparticles left by the arc on surfaces which were in line of sight with thecathode. Macroparticles are discussed in Chapter 6, and here it is only pointedout that size distribution functions are power laws extending over several ordersof magnitude, exhibiting a self-similar (fractal) property: An observer looking atelectron microscope photographs of macroparticles would not be able to decidewhich magnification was used in making the images.

3.5.3 Temporal Self-Similarity

Practically all cathode-spot parameters show fluctuations. Whatever we mea-sure, e.g., voltage, light emission, particle fluxes, and ion charge states distribu-tions, we always see ‘‘noise.’’ One may argue that there are exceptions, forexample the arc current may appear rather constant. This is an artifact of the


circuit that occurs when the circuit impedance is much greater than the impe-

dance of the arc plasma.By measuring spot data with increasingly higher temporal resolution, one

would ‘‘zoom in,’’ just like one could look at coastlines with increasingly higher

spatial resolution. Analogously, one might find fractal features in the time

domain, known from time series analysis. Figure 3.18 shows an example of

time-resolved light emission from a copper vacuum arc. Similarly looking curves

can be obtained with different time resolutions and from different physical

quantities, such as ion current collected by probes or shields.In the late 1980s, Smeets and Schulpen [79] studied the correlation of light

emission and ion current signals for low-current copper arcs. Taking time-of-flight

of ions into account they found that high-frequency fluctuations of voltage and

ion current were correlated, while the average light intensity was proportional to

the arc current. Fast Fourier transform (FFT) of noisy ion current signals did not

reveal distinct peaks, indicating that the measurements did not reach the lower

cutoff. The FFT spectrum showed a 1/f-character for the spectrum up to about

15MHz and approximate ‘‘white’’ noise for faster fluctuations (Figure 3.19).

Fig. 3.18. Example of time-resolved light emission from a copper vacuum arc; the curveis a representative of time-resolved data with fractal character. Similar curves can be

obtained on changing the time resolution [88]

Fig. 3.19. Fast Fourier transform (FFT) of noisy ion current signals [79]; the spectrumdoes not reveal distinct peaks but has 1/f character up to 15MHz


Unfortunately, their display of data was linear and it is hard to judge how well the

noise was indeed represented by a 1=f dependence. The slope of a power law, if

present, is much easier to find when data are displayed in log–log presentation.

Furthermore, they did not explicitly distinguish between Fourier amplitude and

Fourier power spectrum but we may assume that they referred to the amplitude

spectrum. The corresponding power spectrum would be 1=f with � 2, i.e.,

brown noise.Usingmore recent data [91], random color noise (RCN, [84]) was found within

the physically meaningful frequency interval. Most data curves show three

characteristic segments. There was white noise at high frequencies. At inter-

mediate frequencies, a RCN slope of41 was found.Many curves had a kink at

lower frequencies, giving 0551, which should be disregarded, as discussed

below. Figure 3.20 shows such curve for the noise of Cu ions for a pulsed vacuum

arc.White noise at high frequencies is possibly due to ion velocitymixing [91], that

is, all information from emission sites is mixed because faster ions overtook

slower ions on their way from the cathode to the ion collector. ‘‘Scrambled’’

information would result in white noise. In the case of carbon, Figure 3.21, the

segment of white noise is particularly pronounced. Carbon ions are lighter and

faster than other metal ions, and the effect of velocity mixing could be more

pronounced. However, research is still ongoing; white noise can also be caused

bymeasurement limitations, in particular when the amplitude of the noisy signal

approaches the noise of the measuring system itself.At low frequencies, the curve is quite uncertain due to insufficient data.

Mathematically, because we deal with a finite discrete sample of data, the limits

Fig. 3.20. FFT of ion current for a copper vacuum arc pulse. One can see the approximatepower law in the physically relevant interval. (From [3])


of discrete Fourier transforms (DFT) apply: The inverse of the sampling period

represents a lower limit to the resolution of the transformed data, and the upper

limit is half of the sampling interval (Nyquist sampling theorem). Only few data

points contribute to the lower frequency part of the FFT curve, and therefore it

should be disregarded.More research is needed to really learn what information can be extracted

from the RCN segment of such FFT curves. So far, one can see that the relevant

segment appears close to brown noise, an indication for the relation to Brownian

motion of spot ignition. One should note that the greater the color coefficient the more one is tempted to search for ‘‘characteristic’’ frequencies or time

intervals. For example, emitted light appears to have same smaller peaks and

valleys and larger peaks and valleys suggesting periodicity [88, 92]. Beilis and co-

workers [92] identified brightness fluctuations for copper arcs with intervals of

17–3 ms. However, Fourier analysis with sufficient data is needed to determine

whether the peaks and valleys are periodic or stochastic.The arc voltage is an especially suitable parameter to research cathode

processes because it reflects them in a less distorted way than ion current or

most other parameters. This is because phase mixing, as it is prevalent with ion

currents [91], or ‘‘smearing out’’ of emitted light due to finite, non-zero lifetime of

excited states [88] does not play a role. Using voltage noise, the signal generated

by the cathode processes can be recorded with maximum fidelity, only limited by

the bandwidth of the recording circuit.Using a coaxial arc discharge arrangement and broad-band voltage divider

and attenuator measurements, the FFT of arc burning voltage was determined

Fig. 3.21. FFTof ion current for a carbon vacuumarc pulse, which is similar to the copper

case, Figure 3.2, but white noise extends to lower frequencies. (From [3])


with maximum resolution possible [93]. Figure 3.22 shows the result for eight

cathode elements. As with ion current, one can recognize the 1/f 2 character in

the power spectrum, i.e., the voltage noise has fractal character which can be

associated with Brownian motion of spots (see next section). The differences

between materials will be considered in Section 3.7.

3.5.4 Fractal Character and Ignition of Emission Centers

While fractal character results from many stochastic events, research of ele-

mentary processes requires reducing (avoiding) the superposition of spot

fragments and ignition events, and hence research was done with the smallest

current possible. In a careful study with very small arc currents, Puchkarev and

Murzakaev [94] found a correlation between voltage fluctuation and appear-

ance of erosion craters. Figure 3.23a shows a track of erosion craters with a gap

between one crater and a chain of craters. In this particular event, the gap

could be associated with a spike in voltage (Figure 3.23b). The interpretation is

Fig. 3.22. FFT of the burning voltage for eight cathode materials. One can clearly see thepower law describing brown noise (spectral power 1/f 2 ) (after [215]). The FFT of themeasuring system noise (no arc) is also displayed


as follows: After an explosive event, which created the crater, the emission

center ceased to produce plasma, the impedance of the discharge increased,

causing the voltage to rise up to the full applied voltage. One should note how

remarkably fast the production of plasma ceased. At this point, the arc might

have become extinguished (‘‘chopped’’). However, the increase in voltage

increased the field strength on the surface, which in turn increased the energy

of returning ions through acceleration in the cathode sheath, as well as emis-

sion of electrons, as explained at the beginning of this chapter. New emission

centers were ignited, and in this example the sequence of further ignition events

was rapid and practically uninterrupted.This study supports what is intuitively clear: Ignition and ‘‘death’’ of emission

centers are related to voltage fluctuations, and ignition is affected by electric

field strength and ion bombardment heating. The momentary and local field

strength results from the combination of momentary voltage, local sheath

thickness (plasma density), and local field enhancement factors. Even if we

have for a moment a homogenous sheath, the cathode–sheath–plasma system

is not stable against changes caused by a local enhancement of electron emission

[95]; the electrons have a nonlinear feedback on local plasma density, sheath

thickness, and emission current.

Fig. 3.23. Track of erosion craters with a gap between one crater and a chain of craters(top). In this particular event, the gap could be associated with a spike in voltage (bottom).(After Puchkarev and Murzakaev [94])


Suitable ignition sites may be distributed randomly or evenly over the surfacebut the distribution of actual ignition events may be modeled using DLA-likeapproach. In a first approximation, one may apply a simple stochastic DLAmodel which was originally developed by Niemeyer and co-workers [96] todescribe the formation of Lichtenberg figures. As illustrated in Figure 3.24,dark spots correspond to locations with plasma, having potential � ¼ 0, sur-rounded by locations of potential � ¼ 1. The initial location of ignition is in thecenter. Suppose the current is high enough to require several emission centers tobe active. They correspond to the tips of the branches coming from the center.Empty circles indicate the locations where a new emission center could beignited. The dashed lines between full circles and empty circles symbolize pos-sible bonds in the diffusion-limited aggregation or steps in Lichtenberg figuresor arc spot displacement. A probability can be assigned to each of those dashedlines expressed by

p i; k! i0; k0ð Þ ¼�i0;k0� �P

�i0;k0� � ; (3:72)

where the exponent is introduced to adjust the relation between local fieldand probability. For ¼ 1, the growth probability is directly proportional tothe local field, resulting in a fractal with Hausdorff dimension of about 1.75,which is comparable with Lichtenberg figures (Figure 3.16) or arc traces ofspots (Figure 3.25).

Fig. 3.24. A stochastic DLA model for spot ignition, originally developed by Niemeyerand co-workers [96] to describe the formation of Lichtenberg figures. The dark spots atthe end of branches indicate active spots, while empty spots correspond to the potential

ignition sites


The model above introduces an important element to a more comprehensivespot model: it conceptually connects the distribution of the ignition probabilitywith the local field distribution, which in turn resulted from previous ignitionevents. This recursive relation to previous behavior is the basis for a fractal: theresult of the next step of development depends on the immediate previouscondition, which will be mentioned again as a Markov process in the discussionof apparent spot motion. It will be argued that cathode-spot types, apparentspot velocity, retrograde motion, and other spot effects can be described by astochastic fractal model for spot ignition.

Picosecond laser interferometry and absorption shadow imaging were usedby Batrakov and co-workers [97] to study plasma formation on a liquid metalcathode. Using relatively small currents of less than 50 A, they observed spotfragments and determined the plasma density to be of the order of 1026 m–3. Thesame group investigated a capillary-type cathode of gallium–indium alloy usingresonant laser absorption imaging with the spectral lines of neutral vapor,finding that fragments operate in a cyclic manner [98]. Such cyclic processeswould constitute the small-scale, fast-process physical cutoff for a fractal spot.

At even lower current, about 10 A, less emission sites are simultaneouslyactive. Therefore, the ignition of fragments and arc instabilities are best inves-tigated at low arc currents. Low current conditions were selected by Tsuruta andco-workers [99] showing that the voltage rises when the current drops, as onewould expect from models based on ignition of emission sites. Consistently, anaxial magnetic field does not help to stabilize the arc, rather makes it moreunstable, especially when the anode becomes magnetically insulated, i.e., whenelectrons need to cross magnetic field lines on their way from cathode to anode.

Fig. 3.25. Traces of arc spots observed on an oxide-coated metallic shield. Note thesimilarity between Lichtenberg figures (like in Figure 3.16) and DLA clusters (likeFigure 3.24). The Hausdorff dimension is about 1.7


Explosive emission events excite a broad range of soundwaves that propagatein the solid cathode body and its surface. Techniques of acoustic surface wavesanalysis are often used to determine Young’s modulus of coatings [100], and theycan also be used to learn about the processes at cathode spots. Laux and Pursch[101] studied sound waves on graphite, carbon fiber reinforced carbon, andstainless steel cathodes. They detected distinct frequencies in the power spec-trum. Most of the sound modes showed strong damping demonstrating that thenonlinear disturbance associated with the initial breakdown excited a widespectrum. They modified the electrode geometry to identify eigen-oscillationsof the cathode plate. So far, it was concluded that different disturbances duringthe lifetime of the moving arc are reflected in different selections of sound eigen-modes. This approach might deliver interesting information when coupled toother diagnostic techniques.

3.5.5 Spots, Cells, Fragments: What Is a Spot, After All?

It was mentioned that the appearance of spots depends on the resolution of theequipment used for observation. Every time equipment of higher spatial andtemporal resolution became available, new smaller and faster structures weredetected. This is particularly evident in the still ongoing debate on the currentdensity of cathode spots. In the framework of a fractal spot model, one has torecognize that there is no single value, or even single peak value, of currentdensity. Figure 3.26 illustrates the debate on current density over the decades;the presentation was originally shown with a wink in the 1980s but can now be

Fig. 3.26. Measurements of current densities of cathode spots: different measuring app-roaches gave different results, and with the availability of higher resolution equipment,smaller structures (hence higher current densities) were claimed. The debate loses most ofits contention when the spot is seen as a fractal object


reinterpreted via the fractal approach. In this sense, the quest for the current

density is comparable with the now-famous quest for the ‘‘true’’ length of

coastlines.Cathode spots are often discussed as if they were circular with a uniform or

peaked current density. As early as in the 1960, Kesaev [57] postulated the exis-

tence of a substructure of ‘‘cells’’ within each cathode spot. Harris [102] considered

cells to explain the retrograde motion of cathode spot. Modern, high-resolution

imaging techniques such as pulsed laser absorption photography [103, 104],

pulsed laser interferometry [105], and high-speed image-converter photography

[90, 106–108] confirmed a substructure within each spot. Much research has

focused on finding the smallest spatial structures and fastest temporal events

below which magnifying or ‘‘zooming’’ would not reveal self-similar structures.

Many researchers believe that at the scale of crater dimensions and nanosecond

explosions, the lower cutoff of self-similar scaling has been reached. That,

however, remains to be shown by future research. So far, ever smaller structures

have been identified when increasing the resolution [109]. Well-defined plasma

jets ejected by plasma instabilities from within a cathode spot [110] indicate that

even finer structures may exist at even shorter timescales (see also section on

retrograde motion). These smallest units of explosive events would represent

ectons, which were introduced before.These arguments are supported by the highest resolution pictures available to

date. Figure 3.27 shows a cathode spot consisting of two fragments imaged for

10 ns by an image-converter camera [108]. The highest resolution photographs

indicate that each of the fragments may have a substructure or they are highly

Fig. 3.27. Image-converter photograph of a cathodic arc type 2 spot on copper at a currentof 30 A. The spot consists of two fragments. The short exposure time of 10 ns suggests that

the two fragments exist simultaneously, rather than subsequently. Their irregular form isindicative for either a further, finer substructure or dynamics occurring within the 10-nsexposure. (From [108], Figure 4.13)


dynamic emission centers, showing local displacement within nanoseconds. Thiskind of imaging reaches the physical limits that are given by the finite (non-zero)lifetime of excited levels. The emission of photons that are registered with theimage-converter camera result from fast moving atoms and ions, and the loca-tion of photon emission (de-excitation) is shifted with respect to the location ofexcitation. To circumvent this limitation, absorption techniques using shortlaser pulses can be used. Figure 3.28 is an example of a laser absorption photo-graph of a cathodic arc type 2 spot on copper for a discharge current of 90 Awithan exposure time of 400 ps [103]. It shows a diverse distribution of densities in aregion that one would characterize as a single spot if judged by light emission.Similar results were obtained using even shorter ‘‘exposure time’’ of 100 ps inlaser absorption photography [70].

The simple question ‘‘What is a cathode spot?’’ does not have a simple answerbut the definition ‘‘A cathode spot is an assembly of emission centers showingfractal properties in spatial and temporal dimensions’’ captures the essentialelements. The terms spot fragment, cell, and emission center are used synony-mously. Each ignition event and resulting emission center may be described asan elementary step, corresponding to a sequence of emission stages, whichincludes the explosive or ‘‘ecton’’ stage in Mesyats’ framework. The assemblyof fragments exhibits fractal properties, and the individual steps are the small-scale, short-time cutoffs of spatial and temporal self-similarity.

3.5.6 Cathode Spots of Types 1 and 2

Cathode-spot types 1 and 2 have already been introduced in Sections 3.4.8 and3.4.9. Here we return to this topic because the surface effects are among the most

Fig. 3.28. Laser absorption photograph of a cathodic arc type 2 spot on copper at a

current of 90 A, clearly showing substructure of what normally appears as a singlespot. The laser pulse length (picture exposure time) was 400 ps. The gray scalerepresents the degree of laser light absorption (originally color coded). The electrodeshape was added by image processing. (From [103])


critical for cathodic arcs, and their role has been re-discovered several times.

Apparently, they deserve special attention and need to be brought in perspective

for the fractal approach to arc spots.Systematical studies started in the middle of the twentieth century trying to

nail down the links between phenomenological appearance and underlying

microscopic processes. Among the first reported observations where those by

Dryvesteyn, Suits, Hocker, and Cobine. Dryvesteyn [111] realized that the

formation of an insulating film, such as an oxide, allows charge to accumulate

on the surface, thereby enhancing the electric field strength. Suits and Hocker

[112] and Cobine [113] investigated the glow-to-arc transition. They found a

large element of randomness on the one hand but a clear influence by the

presence or absence of an oxide layer. Looking at relatively small arc currents

in the range 1–10 A on a copper cathode, reduction of the oxide layer with

hydrogen plasma forced the random arc to return to a high-current glow mode

[112]. The observation that either the arc or glow mode is preferred, depending

on the surface conditions, was also confirmed for other cathode materials, such

as Cd, Fe, Al, and Zn [113].The distinction between cathode-spot types 1 and 2was introduced in the

1970s when the role of surface contamination and surface condition on spot

formation and operation was systematically investigated [114]. The availability

of electron microscopes for the investigation of arc traces was certainly an

important factor. The association with numbers ‘‘1’’ and ‘‘2’’ can easily be

remembered by recalling that the erosive action of the cathode spot effectively

cleans the cathode surface. Spot operation starts with type ‘‘1’’ on a contami-

nated surface, which is initially always present (unless the system was treated in

ultrahigh vacuum), and it switches to type 2when surface contaminations are

removed by action of spot operation.There are numerous phenomenological differences between type 1 and 2

spots, as becomes obvious by in situ optical and electrical observations as well

as by postmortem examination of the cathode in an electron microscope [65, 80,

115]. Spot type 1 appears dim compared to type 2; the velocity of the plasma

front and the apparent spot motion are much greater for type 1. The arc voltage

and cathode material erosion is smaller for type 1. The average current per spot

is much smaller for type 1 spots; at small current, spot type 2may not operate at

all. This also explains why the discharge went to the glow mode rather than

transitioned into type 2 in the 1930s’ experiments mentioned before [112, 113].

When the cathode is examined in an electron microscope one clearly recognizes

that craters left by type 1 spots are much smaller than craters of spot type 2 [65,

116]. Spot type 1 craters are isolated and spread apart by many crater diameters,

whereas spot type 2 craters are larger and next to each other, forming character-

istic chains of craters [117]. In the presence of an external magnetic field, arc

craters are aligned in a band of well-separated craters for type 1, and in amore or

less linear chain of craters for type 2, as illustrated in Figure 3.14. Table 3.5

summarizes some phenomenological differences between type 1 and 2 spots.


Cathodic arc investigations before the 1960s were usually done with type 1spots because typical vacuum conditions did not allow for clean, metallicsurfaces. However, since the arc spots remove material hence performingin situ cleaning, it is possible, or even likely, that transitions to type 2 spotsoccurred when the arc duration was long and the vacuum conditions werereasonable. Results of early work must therefore be judged with some cautionsince the type of operation may not be clear. In the newer literature, research-ers have become aware of the importance of surface conditions, and often thetype is clearly defined [115, 118]. In fact, most cathode-spot research after1980 focused on spot type 2 observed on clean surfaces, because only herereproducible results could be obtained. In the other case, the nature of thesurface contamination or coverage needed to be characterized, which madethe task much more difficult. Worse, even if one was successful in character-izing the cathode surface conditions prior to the occurrence of a cathodic arc,the erosive cleaning action will change the surface rapidly in a manner thatwill depend on numerous factors, some of them difficult to control. Amongthese factors are the residual gas pressure and gas species, pumping speed,nature of cathode material, and cathode temperature. Great progress has beenmade in measurement and modeling of cathode spots of type 2 because theexistence conditions are well defined.

For spots of type 1, relatively little has been done because the task is dauntingdue to the large variety of surface conditions possible. For the purpose ofcathodic arc coatings, reactive gases (often nitrogen or oxygen) are commonlyintroduced in the deposition chamber, and the measurements and models forspot type 2 are only of limited value. In practical coating systems, containing

Table 3.5. Phenomenological, qualitative differences between type 1 and 2 spots. Allstatements should be seen as relative for a given cathode material; due to large variations

between cathode materials, absolute values are not given here

Type 1 Type 2

Surface conditions Contaminated (i.e., oxide) Clean (metal)

Apparent brightness Dim BrightTypical crater diameter Small LargeCrater appearance Separate from each other Adjacent to each other

Apparent spot velocity High LowCathode erosion rate Low HighRelative ease of arc triggering

and burning

Easy Difficult

Chopping current Low HighPlasma composition Metal and gas (hydrogen,

oxygen, etc.)

Metal only

Average ion charge state Low HighRelative amplitude offluctuations

Low High


reactive gas, arc spotsmost likely work primarily via type 1 or amixture of type 1

and 2 spots. Much research remains to be done to clarify spot processes under

the conditions of reactive deposition. Szente and co-workers [119–121] have

done pioneering research in this direction. Among other results, they found

that the cathodic arc on cleaned copper electrodes operates with spot type

2when high-purity argon is present; however, the copper erosion rate was

drastically reduced by the addition of only 1% of nitrogen and was further

reduced as the nitrogen content increased in the gas mixture. The decrease was

found to be correlated to an increase in arc velocity [119], clearly indicating a

change to type 1 spots.

3.5.7 Cathode Spots on Semiconductors and Semi-metals: Type 3

Cathodic arcs require that the cathode has sufficient conductivity to carry the

arc current, which is often 50 A and higher. If the conductivity is too low, high

voltage drops and associated high ohmic losses would occur in the cathode.

Therefore, as a rule, cathodic arcs are limited to metals. A number of sufficiently

conducting semiconductors and semi-metals have successfully been used, includ-

ing highly doped Si and Ge, graphite, hot boron, and boron carbide. The

appearance and mechanisms of cathode spots on these materials are modified

compared to operation on true metals.One important phenomenological difference is in the apparent spot velocity,

which is much smaller than onmetals; in fact, the arc spot has a tendency to stick

to the same location for considerable time. Using a far-distance microscope,

Laux and co-workers [122] observed this spot behavior in situ on B4C, and the

observation is supported by cathode erosion profiles. Experiments with a gra-

phite cathode often show deep holes that relatively stationary spots have ‘‘dug’’

into the cathode.It seems to be clear that the temperature dependence of the electrical con-

ductivity plays an important role since, in contrast to ordinary metals,

d�

dT50; (3:73)

i.e., the higher the local temperature the lower the local resistivity. While this

may contribute to stabilizing the spot, thermal conductivity will also occur here,

increasing the spot area and decreasing the power density until it is insufficient

for cathodic arc operation. Because the appearance of spots on semiconductors

and semi-metals is quite different compared to type 1 and 2 spots on metals, one

may assign the term ‘‘type 3’’ to arc spots on non-metals.Using his model of cathode layers, Beilis [123] formulated a system of time-

dependent equations and applied it specifically to graphite. Spot parameters

varied strongly when the spot lifetime was assumed to be shorter than 10 ms. Inthe case of graphite, Joule heating in the cathode body is significant and may


exceed cathode heating by the ion heat flux, which is contrary to conclusions for

metals found by Juttner [107].The physical properties of graphite can vary widely depending on the manu-

facturing procedures. Kandah and Meunier [124] found that the cathode-spot

velocity is higher on graphite of large grain size, low electrical resistivity and high

density, small pore size and less porosity.Arc spots on hot boron (heated up to 1,0008C) have been investigated by

Richter and co-workers [125] using fast framing photography, mass spectro-

scopy, and ion energy analysis. They detected an extended molten area on the

cathode surface in which cathode spots move. The ions emitted were B+ and

B2+ with kinetic energies up to 90 eV, indicating that the fundamental character

of the emission sites is cathodic, as opposed to thermionic, although the spot

motion is distinctly different (slower) than on metals.

3.5.8 Arc Chopping and Spot Splitting

Many experiments have shown that there exists a minimum arc current needed

for stable, self-sustained arc operation. Discharge currents lower than what is

called the ‘‘chopping current’’ will lead to spontaneous extinction of the arc.

A minimum current is required to ensure sufficient plasma production, which

ensures a very high likelihood of ignition of a new cathode emission center when

the power density at the active center has dropped. Such ‘‘minimum arc main-

tenance value’’ was already known in the early twentieth century for the con-

tinuous operation ofmercury arc lamps [126]. This minimum current depends on

the cathode material as well as on the surface state of the cathode. For example,

the chopping current for pure titanium is about 50 A [127]. Electrode materials

developed for vacuum arc switches, such as CuCr, CuW, and AgWC, show very

small chopping currents of order 1 A [128], which is important in order to

minimize the induced voltage. High induced voltage could cause re-ignition of

an arc and failure of the switching event.Going in the other direction of higher currents, it was found that the amount

of plasma produced is directly proportional to the arc discharge current. Inter-

estingly, other parameters such as burning voltage, average ion velocity, and

mean ion charge state show only small changes with increasing arc current.6

This may be surprising at first sight but it becomes plausible if one considers that

higher currents lead to ‘‘spot splitting.’’ Spot splitting means that a larger number

of simultaneously active emission sites exist, proportional to the arc current,

where each of the spots maintains approximately the same operational mechan-

ism and carries about the same average current [92]. The literature, inmost cases,

6 This is not true for very high currents, e.g., arcs with currents in the kiloampere region,when anode activity occurs and when the magnetic self-field cannot be neglected.


refers here to current per ‘‘spot’’ and it is likely that each spot is an assembly offragments or emission sites, as discussed before.

3.5.9 Random Walk

The spot plasma or the spot ‘‘itself’’ does not move on the surface but it is thelocation of ignition that moves. Therefore, although the term spot motion isintuitive and reflects a phenomenon that can be visually observed, it is ratherabstract because this motion should always be understood as a sequence ofignition and extinction of active emission sites.

Random walk [129] is generally introduced by assuming that a ‘‘particle’’ (thespot in our case) starts at the origin (the location of first ignition) and takes a stepof length s in a random direction specified by the vector n, with nj j ¼ 1. Theposition after N steps is RN. In the next step,

RNþ1 ¼ RN þ sn; (3:74)

with R0 ¼ 0. Taking the average over many trajectories,

RNþ1h i ¼ RNh i þ s nh i ¼ RNh i (3:75)

because nh i ¼ 0. This leads to

RNh i ¼ R0h i ¼ 0: (3:76)

This result was expected because there is no preferred direction, therefore theaverage over many test walks of our ‘‘particle’’ must be the starting point.

Looking at a trajectory of a walk after N+1 steps, the spot has reached adistance squared from the origin

RNþ1j j2¼ RNj j2þs2 þ 2snRN (3:77)

and averaged over many trajectories,

RNþ1j j2D E

¼ RNj j2D E

þ s2 þ 2s nRNh i: (3:78)

The directional unit vector n is not correlated with RN and thereforenRNh i ¼ nh i RNh i, which together with nh i ¼ 0 turns (3.78) into

RNþ1j j2D E

¼ RNj j2D E

þ s2 (3:79)

and with R0 ¼ 0 one obtains

RNj j2D E

¼ Ns2: (3:80)

Thorough investigations by Schmidt [130], Daalder [78], and Hantzsche andco-workers [77] established that in the absence of an external magnetic field, spotmotion may be modeled as random walk. One may assume that each displace-ment occurs in a small elementary step, s, which takes an average elementarytime t.When considering a two-dimensional randomwalk (i.e., on a surface), the


probability P(R) for a total spot displacement to be in the interval (R, R+dR),

with R ¼ Rj j, measured from the starting point, is given by [131]

P Rð ÞdR ¼ R

2Dtexp � R2

4Dt

� �dR for t44t: (3:81)

The functionP(R) is also known as Raleigh distribution. The diffusion constant,

D, contains the parameters of the elementary step:

D ¼ 1

4

s2

t: (3:82)

The diffusion constant is material dependent, for example Juttner found experi-

mentally D ¼ 2:3 0:6ð Þ � 10�3 m2=s for copper [107] and D � 10�3 m2=sfor molybdenum [132], and Beilis and co-workers [92] determined

1 0:3ð Þ � 10�3 m2=s for copper and 4 1ð Þ � 10�4 m2=s for CuCr contact

material.The mean value for displacement is

Rh i ¼ð1

0

R P Rð Þ dR ¼ pDtð Þ1=2¼ s

2

ffiffiffiffiffiffi

pt

t

r

; (3:83)

and the observable, apparent spot velocity is

vspot ¼d Rh idt¼ 1

2

ffiffiffiffiffiffiffipDt

r

¼ s

4

ffiffiffiffiptt

r: (3:84)

From (3.84) one can see that the spot velocity decreases as the observation time

increases, which is a consequence of the random nature of this motion.At this point it should again be emphasized that this motion is apparent and a

velocity is only defined in the sense of ‘‘smearing out’’ elementary steps and

considering changes of R for long observation times: t44t. Strictly speaking,

elementary steps are associated with ignition events, and therefore this ‘‘motion’’

does not have a derivative, hence a velocity in the usual sense is not defined.The assumption of an elementary step s deserves further critical review. Based

on analysis of arc traces on clean metal surfaces (spot type 2), Hantzsche and co-

workers [77] suggested using the mean crater radius as the elementary step

length. The assumption is justified by the appearance of crater chains, though

its general validity was disputed byDaalder [133]. He also questioned whether or

not there are two distinct cathode mechanisms, one producing craters adjacent

to each other, associated with spot type 2, and another producing clearly

separated caters, typical for type 1. Using the fractal approach to spot phenom-

ena, this dispute loses its relevance because the issue is reduced to a probabilistic

distribution of ignition of emission centers. In this picture, the step length is not a

constant, as assumed in a simple random walk model. In particular, the dis-

tances between ignition locations (corresponding to grid nodes in a randomwalk

model) can be large or very small.


The displacement dynamics of random walk can be modeled using a prob-abilistic approach with application of simple ad hoc displacement rules forignition of new emission centers on a two-dimensional grid. Such a model wasdeveloped by Coulombe [134] for high-pressure arcs, though some of the findingsare applicable to vacuum arcs as well. The model predicts a power law signatureof the frequency power spectrum, (3.71), where the exponent depends on thelarge-jump probability. For the condition simulated [134], the exponent variedfrom ¼ 0 for smooth diffusion conditions (no jumps to distant grid nodes)to ¼ 1:6 for a 0.9 likelihood of ignition far from the active emission site.The appearance of a power law is again a signature of fractal behavior.

Analyzing the 50-A arc traces on an aluminum film, which are qualitativelysimilar to arc traces on oxidized surfaces (like Figure 3.25) or Lichtenbergfigures, Anders and Juttner [80] found a fractal dimension of 2within 5%,which is the expected dimension for two-dimensional Brownian motion.

The well-known Brownian motion is the scaling limit of random walk. Thismeans that if random walk occurs with very small steps, s! 0, random walkbecomes an approximation to Brownian motion. Often, to make modeling andcomputation more efficient, Brownian motion is approximated by a randomwalk. Random walk is a discrete fractal exhibiting stochastic self-similarity onlarge scales, but self-similarity is cut off as scales approach the elementarystep width s. Brownian motion in two dimensions is a true fractal showing self-similarity on all scales. Brownianmotion has the fractal (Hausdorff) dimension 2.

By superimposing a narrow Gaussian distribution of light intensity on a plasmaemission center making a random walk, it was possible to simulate the lightemission patterns observed from cathode spots [88]. A physical interpretation fora Gaussian distribution is based on the finite (non-zero!) lifetime of excited levels ofatoms and ions. The location and size of the rapidly expanding plasma of anemission site is observed by photons, many of them originating from bound–boundtransition of ions and atoms (e.g., for copper [135]). The longer the lifetime ofexcited levels, which is especially important for long-living metastable levels, thefurther the distance between excitation near the spot center and de-excitation in theexpanding plasma, and therefore the broader the Gaussian distribution.

3.5.10 Self-Interacting Random Walks

In a random walk, the next position depends only on the current position of the‘‘walker’’ (the emission site). The random walk has no memory of all otherprevious positions, i.e., walk history. Such process is called a first-orderMarkovprocess [76]. Provided that cathodic arc processes occur on clean metal surfaces,such as well-arced cathodes, all sites next to the active site have approximatelythe same probability to serve as the next emission site, and the walk may bemodeled as a Markov process.

More realistic, however, is that the arc changed the surface conditionsand local temperature in such a way that the distribution of the ignition


probability is changed. Experiments (like the breakdown experiments by Kraft

and Stuchenkov [136]) show that arcing on contaminated surfaces is relatively

easy, while arcing on metallic, clean surfaces is relatively difficult. Therefore,

the cleaning action of arc erosion will markedly reduce the ignition probability

Fig. 3.29. Two examples of observation of the memory effect using high-speed photo-graphy on a low-current copper arc. One should compare first and last frames of the leftand right sequence, which shows that the spot returns to emission sites previously used.

(Courtesy of B. Juttner, [140])


of locations where the arc has already burned. Starting with an oxidized orotherwise ‘‘contaminated’’ cathode, the random walk will turn into in a specialclass of self-interacting walk, the self-avoiding random walk [137, 138]. As aresult of arc cleaning, the next emission is much more likely to be ignited atlocations that still have the surface-field-enhancing oxide layer rather than onthe arc-cleaned surface. As long as there is surface area with oxide layer (orsimilar contamination) available, emission sites will avoid the cleaned surfaces.This feature has been utilized in arc cleaning of steel surfaces where oxidesurfaces were intentionally created to optimize cleaning effects [139].

If arcing occurs in a reactive gas environment or poor vacuum, the surfacewill quickly ‘‘age’’ by becoming ‘‘poisoned,’’ i.e., the metal atoms on the cathodesurface will react with oxygen or water from the residual gas. In this way, thesurface will ‘‘forget’’ its memory on the local emission event, and the walk canagain become Markovian rather than self-avoiding.

If arcing of the surface occurs in ultrahigh vacuum (UHV), or in noble gas freeof reactive components, prolonged arcing may eventually completely remove theoxide layer. As a consequence, all surface locations become equivalent, and also inthis case the self-avoiding walk will switch back to a random (Markovian) walk.

Therefore, in both extreme cases, arcing with reactive gas and arcing in UHV,there is a tendency to go from random walk through a phase of self-avoidingwalk, which eventually may be followed by another phase of random walk.

The picture, however, is evenmore complicated.Not all self-interacting randomwalks are self-avoiding walks [138]. For example, investigating clean, well-arcedcathodes, it has been observed that after a short cooldown time, a previously activeand still-hot emission site may preferably re-ignite [140]. In this case, the walk isnot self-avoiding but rather shows a preference to return to a familiar location(Figure 3.29). Juttner’s experiments showed that ignition of arcs on hot cathodesrequire higher ignition voltages than on cold cathodes, which fits with the picturethat emission centers appear on new locations rather than on the already hot sites.Therefore, other factors must play a role for the memory effect. The roughstructures on a rim of cathode craters seem to play a role: in a photo series withvery high resolution, Juttner and Kleberg succeeded in observing the ‘‘dance’’ ofignition around the sharp asperities of a large crater (Figures 3.30 and 3.31).

3.5.11 Steered Walk: Retrograde Spot Motion

In the presence of a transverse magnetic field, the ‘‘motion’’ of ignition locationsdeviates from random. Rather, this apparent or virtual motion becomes increas-ingly directed with increasing field, i.e., the ignition of emission centers is morelikely in a preferred direction. Spot motion is magnetically ‘‘steered.’’ The term‘‘steered arc’’ is often used by the arc coating community. The preferred directionof the virtual motion is opposite to the Amperian direction j� B which onewould expect if the Lorentz force was responsible for motion. The plasmacolumn is indeed bent in the j� B direction, as indicated in Figure 3.32, though


Fig. 3.30. High-resolution imaging of spot fragments igniting preferably on the asperitiesof a large cater. Copper cathode, arc current 30 A, frame exposure time 10 ns, first frame

taken 300.00 ms after arc triggering. (Courtesy of B. Juttner, [140])


the virtual motion is to the opposite (anti-Amperian or ‘‘retrograde’’) direction.

Since its discovery [141, 142], retrograde motion has stimulated much research

[102, 121, 143–178], resulting in a number of more or less convincing explana-

tions. One of the most comprehensive descriptions is by Juttner and Kleberg

[108, 110] who base their model on observations using image-intensified cameras

of very high spatial and temporal resolution. Much of this section is based on

findings by Kleberg [108].

Fig. 3.31. Large crater on a copper cathode, corresponding to fast framing picturesshown in Figure 3.30. (Courtesy of B. Juttner, [140])

Fig. 3.32. Bending of the plasma column in the j� B direction, with spot motion (i.e.,ignition of new emission centers) in anti-Amperian (–j� B) direction


As the magnetic field increases, the virtual motion loses its random

character and becomes increasingly directed, which allows us to assign a

macroscopic velocity rather than diffusion constant [171]. The directed retro-

grade velocity depends on a few conditions, including magnetic field strength

and cathode material. It is typically 10–15m/s for a steering field strength of

about 15mT [179]. The directed velocity is approximately proportional to

the transverse field Bt, i.e., the field component parallel to the cathode

surface,

�steered ¼ c Bt; (3:85)

where c is amaterial constant that also depends on the surface conditions. For clean

surfaces under vacuum conditions, Zabello and co-workers [178] determined c to be

60 and 200m/(s T) for copper and CuCr cathodes, respectively. The parameter c is

smaller when the cathode is hot (4 6008C in the case of copper) [169].The direction reverses into the Amperian when the pressure exceeds a critical

value, which is relatively high, namely in the range 1–100 kPa (i.e., from a few

percent of to about atmospheric pressure) [157, 164].Let us further consider the phenomenology of the vacuum and low-pressure

case. As the magnetic field vector intersects the cathode not normal but tilted, the

direction of the virtual motion is also tilted to the retrograde direction (Figure 3.33).

Fig. 3.33. The direction of the apparent motion is tilted to the direction of retrograde

motion as the magnetic field vector intersects the cathode at an angle other than normal


The angle between retrograde direction and actual direction is often called theRobson angle, after A.E. Robson who worked on the issue with A. von Engel inthe 1950s [153]. Measurements by Zabello and co-workers confirmed that theRobson angle depends directly on the field inclination [177, 178] (Figure 3.34). Asthe temperature of the cathode increases, larger craters are observed and the retro-grade velocity decreases [168, 169]. In an early work of 1948, Smith [147] observed areversal of motion direction but that has not been confirmed; most likely hisobservation was a pressure rather than a temperature effect.

The interpretation and modeling of steered and retrograde motion has beenfound to be difficult. It is associated with the ignition probability distribution. Inthe absence of an external field, the magnetic self-field is azimuthal and thusaxially symmetric, provided the discharge geometry, like position of the anode,does not break the symmetry. If themagnetic field has a component parallel to thecathode surface (often called the transverse or tangential field), the axial symmetryis broken. Putting the physical reason for preferred spot ignition aside, for the timebeing, one could use probabilistic models for spot motion, in extension to pre-viouslymentioned randomwalkmodels. Care [180] constructed aMarkov processon a two-dimensional grid where the probabilities for the directions are not equalbut depend on the steering magnetic field. If the grid size and time intervals areallowed to go to zero, and appropriate limits are taken, a partial differentialequation for the probability density of the arc position is obtained, which isequivalent to a Fokker–Planck equation for the system.

One of the difficulties is associated with the fact that one would expect theignition probability higher on the side to which the plasma column is bent(Figure 3.32) and where therefore ion bombardment should be higher [169];however, ignition is more likely in the opposite direction.

Modeling of retrograde motion can be done making more or less specificassumptions. For example, Beilis [176] applied his cathode layer model and

Fig. 3.34. Experimental observation of steered spot motion on copper in (left) a pure

transverse field and (right) a tilted field, i.e., where both normal and transverse magneticfield components exist. The transverse field was 65mT in both cases, and the normalcomponent was 60mT for the tilted case (right). (Adapted from Figure 6 of [178])


assumed that (i) heat loss in the cathode bulk is small compared to the heat

influx, (ii) the plasma flow in the Knudsen layer is impeded, and (iii) the plasma

kinetic pressure is comparable to the self-magnetic pressure in the acceleration

region of the cathode plasma jet. With these assumptions, a number of features

can be reproduced, like the linear increase of apparent spot velocity with

increasing magnetic field strength and arc current.The electric current causes a magnetic self-field, which greatly affects the

motion of emitted electrons. Arapov and Volkov [181] developed a model for

ignition of emission centers in which electrons form a current vortex whose axis is

perpendicular to the surface. The current vortex is shown to be unstable, leading

to the formation of a spatial structure. Similar effects are known from the theory

of self-organization when energy is pumped into a system. Locations of enhanced

power density would promote the ignition of the next emission center.Taking an experimental approach, Kleberg and Juttner [108, 110] used spot

imaging with the highest temporal and spatial resolution available. They dis-

covered that emission sites in transverse magnetic fields emit microscopic

plasma jets whose angular distribution is determined by the transverse field

direction. The jets form in a stage of maximum plasma production. It is

known that current-carrying plasma in a magnetic field is subject to plasma

instabilities [182]. Although the exact nature of the instability is still subject to

research, it has been found that the ejection of plasma jets occurs every few

microseconds in directions that deviate from the retrograde direction up to 458.Figure 3.35 shows the ejection of two jets into approximate retrograde direction,

which is toward the right in this photo. Using the sophisticated exposure features

Fig. 3.35. Ejection of two jets into approximate retrograde direction. The intensity oflight emission of the plasma jets is weaker than the emission from the spot itself, and

therefore the exposure was selected such as to make the jets visible, while the cathodespot is overexposed. (Adapted from [108], Abb.4.17)


of the image-converter camera, it was possible to superimpose four exposures

separated by 10 ms. The result is shown in Figure 3.36, which illustrates the

relationship of microjets and ignition of new emission sites. The first spot is at

the left, and retrograde motion is to the right. This picture leads to the Juttner–

Kleberg model of retrograde motion (Figure 3.37): retrograde motion is actually

composed of a microscopic ‘‘zigzag’’ ignition of new emission sites caused by

microjets. The microjets enhance the local electric field strength by E ¼ vjet�B.The local electric field strength was previously identified as a critical parameter

for ignition. The additional field component can be estimated by the measured

jet velocity ( 5 � 103 m/s) and applied field ( 0.2 T). Although it is found

to be only of the order 1 kV/m, this field may be crucial for ion motion and

Fig. 3.36. Superposition of four exposures of 200 ns, each, separated by 10 ms. (Adaptedfrom [108], Abb.4.20)

Fig. 3.37. Juttner–Kleberg model of retrograde motion: retrograde motion is actually com-posed of a microscopic ‘‘zigzag’’ ignition of new emission sites caused by microjets. For low-

current arcs in copper, the typical step length is 200mm. (Adapted from [108], Abb.4.21)


local ion bombardment of the cathode, helping to provide the conditions of siteignition, (3.63).

At a first glance, one is tempted to identify the bright spots in Figures 3.35 and3.36with the emission sites. One needs to recall that these photos were obtainedby intentionally overexposing the spot area to make the less luminous jetsvisible. One should be suspicious because the size of the highly luminous areais much larger than the typical size of craters. If one increases the spatial andtemporal resolution, one finds highly dynamic fluctuations associated withfragments, schematically illustrated in Figure 3.38. Imaging using the emittedlight is reaching the technical and physical limits. To stress the point again: thenon-zero lifetimes of excited states necessarily cause a ‘‘smearing out’’ of theimaged plasma, and therefore the emission sites may be smaller and shorter livedthan suggested by the emitted light.

The Juttner–Kleberg model of retrograde motion reduces the Robson drift toa shift of microjet direction. It was argued that the direction of the microjetejection is determined by the presence of the magnetic field, and it is thereforenot surprising that tilting themagnetic field leads to a shift inmicrojet directions.It is, however, not clear why the zigzag emission appears to have a ‘‘memory’’ ofprevious emissions: an analysis of the motion perpendicular to the averageddirected velocity shows that this component is not random. Rather, the motionis zigzag in a more regular way, allowing the spot to deviate only slightly from

Fig. 3.38. Schematic of fragment dynamics within the Juttner–Kleberg model, as obs-erved when increasing the spatial and temporal resolution. (Adapted from [108],Abb.6.2)


the averaged steered direction. This has been found by Laux and co-workers

[183] on tungsten layers on carbon base material placed in very strong magnetic

fields (a configuration relevant to walls of large thermonuclear fusion devices

such as Tokamaks and Stellarators). It can also be seen on massive cathodes as

shown in Figures 3.14a,b.It should be mentioned that Kleberg found an interesting reversal from

retrograde to Amperian direction before current-zero or the end of a discharge

[108]. Figure 3.39 shows a characteristic U-shape of spot traces, indicating such

reversal of direction.A more practical approach to steered motion will be given in Chapter 5when

discussing the design of arc sources. There we will not consider the physical

reasons based on cathode processes but reduce the apparent motion to two rules:

the retrograde motion rule and the acute angle rule.

3.5.12 But Why Is the Cathode Spot Moving in the First Place?

After this exhaustive discussion on modeling, including fractal description, the

readermay still ask himself/herself: ‘‘But why is the cathode spotmoving in the first

place?’’ First, to stress the point, the cathode spot is not moving. It only appears to

move. What we see is a sequence of ignition and extinction of electron and plasma

emission centers. So, a better question is:Why is there repetitive, stochastic ignition

and extinction, rather than a steady operation (‘‘burning’’) of the cathode spot?Once an emission center is ignited by a thermal runaway process, as explained

before, the conditions for electron emission, plasma generation, and current trans-

fer between cathode bulk and the plasma very quickly deteriorate for three reasons.First, due to the increase in resistance with temperature for all metals,

d�=dT40, the region of the cathode bulk directly under the cathode spot is

more resistive than all other areas or parts of the cathode. Hence, if there was an

Fig. 3.39. Reversal from retrograde to Amperian direction before current-zero or the end

of a discharge. (Adapted from [108], Abb.4.28)


alternative, less resistive way for the current to flow, the current would switch tothe new path.

Second, and this is perhaps the strongest argument, the emission center buildsa highly resistive barrier as the result of the emission. That might sound puzzlingat first. In the explosive stage, the cathode matter transitions from the solid tothe plasma phase, and it may initially bypass the gas phase by circumnavigatingthe critical point in the phase diagram (see Section 3.9). At this stage, the mostresistive zone for the current path is the non-ideal plasma phase. As time passes(and here we consider some tens of nanoseconds), the area of the emission siteincreases by heat conduction, and the area power density falls accordingly,which leads to a change of the path in the phase diagram: the material hasnow the time to transition through all conventional phases: solid–liquid–gas–plasma. From those four, the gas phase is by far the most resistive phase: solidand liquid metal andmetal plasma are good conductors, but metal gas (vapor) isnot. The metal vapor ‘‘chokes’’ the flow of electricity. As the growth of theemission area continues, the power density and related local surface temperatureis reduced, electron emission decreases rapidly, though there may still be sig-nificant evaporation from the hot crater left by the explosion. At this point, thecomposition of the gas or plasma in front of the site becomes increasinglyinfluenced by neutral vapor, and the current transfer capability suffers greatly.

In this situation, the third factor kicks in: competition! The dense plasma nearthe emission site has caused the sheath to be very thin, which implies high electricfield strength on the surface, and the most preferred site may experience arunaway, starting a microexplosion at a new site. Now, the new site and theolder, much larger site are electrically in parallel, and of course the path of lowerresistance (lower metal gas density!) takes over the current.

In this sense, the growing but decaying emission site generates the conditionfor its own ‘‘death.’’ The situation for carbon, boron, and other semi-metals orsemiconductors is somewhat modified in that d�=dT50 and therefore at leastthis reason for the apparent spot motion is removed: quite contrary, thecondition d�=dT50 is stabilizing the spot at one location. Yet, even withthose materials, one can see some (yet slower) apparent spot motion. Thelow-conducting vapor mechanism takes its toll.

3.6 Arc Modes

In the previous sections, cathodic arc spot types 1, 2, and 3were introduced. In thissection, phenomenological spot types are put in relation to arc modes. There areseveral solutions of the energy balance problem and current transport task betweencathode and anode. Depending on the gas type and gas pressure, the role of ioncurrent, ion bombardment heating, thermal properties of the cathode material,cathode surface condition, cathode geometry, and cooling provisions, the solutionsmay be stationary or quasi-stationary for cathodes with relatively large hot areas,leading to thermionic arcs, or non-stationary for globally cold cathodes, leading to


cathodic arcs. Additionally, other components of the discharge circuit may play a

decisive role, like the metal vapor coming from a hot, evaporating anode, leading

to anodic arcs (see below). For a practical means to decide whether a cathodic arc

or thermionic arc is occurring, one should look at the arc burning voltage in a time-

resolved manner, e.g., by using a fast voltage divider probe and an oscilloscope. If

the discharge is burning in the cathodic arc mode one will find the characteristic

noise. In comparison, the noise of thermionic arcs is very small.If one bases arc classification exclusively on the cathode mode of operation,

one may distinguish thermionic arcs and cathodic arcs, with each of them

occurring with different sub-modes and spot types (Figure 3.40). Most of this

book focuses on the cathodic arc mode with spots of types 1 and 2. Thermionic

arcs are only mentioned to clearly define and limit the scope. Thermionic arcs

[47] are widely used in arc discharge lamps [184, 185], plasma torches (e.g., for

plasma spraying [186]), and plasma ion plating systems [187, 188]. Thermionic

modes are governed by the thermionic emission of electrons. There are twomain

sub-modes: the thermionic spot mode and the thermionic spotless mode.In the thermionic spot mode, electron emission occurs via the (field-

enhanced) thermionic emission mechanism, (3.14), from a relatively large hot

spot, which has a diameter of order 1mm or greater. In the thermionic spot

mode, ion bombardment heating and the ion current contribution to the dis-

charge current are crucial. The net ion flow near the cathode is toward the

cathode. The thermionic spot mode occurs, for example, in high-pressure gas

discharge lamps. The thermionic spot mode can be stationary, as it is desired in

commercial discharge lamps [184, 185], otherwise a lamp would be considered

defective. Stationary arc spot models can be used to describe this mode [47, 189].The thermionic spotless mode is similar to the thermionic spot mode but

electron emission occurs from a larger, more diffuse, very hot area that occupies

the entire available working surface of the cathode. To achieve the spotless

mode, the construction of the cathode is usually made in such a way as to

minimize thermal conduction and thermal energy loss by radiation. For exam-

ple, the cathode could be made from a two-segment cathode which has a

cylindrical base of small diameter. The cathode attains a high temperature,

Fig. 3.40. Classification of cathode modes based on electron emission mechanisms

3.6 Arc Modes 147

which is sufficiently high for significant thermionic emission. The thermionic

spotless mode is achievedwhen thermionic electron emission current approaches

the discharge current. At this point, the ion contribution, which is crucial for

current balance in the thermionic spot mode, is not critical to satisfy current

continuity because electron emission is large enough. However, ion bombard-

ment heating may still be critical to maintain the necessary high cathode tem-

perature. The transition from the thermionic spot mode to the spotless mode is

usually sudden, and associated with a slight reduction of the burning voltage.

From the minimum energy principle, the spotless mode is preferred because less

energy is needed to operate the discharge in the spotless mode at the discharge

current determined by the impedance of the external circuit.After the thermionic spotless mode has been established, it may be stable or,

after a thermal time constant which is often on the order of seconds ofminutes, it

may switch back into the thermionic spot mode due to the lack of sufficient ion

heating. The spotless mode is only stable if the cathode energy balance allows the

discharge to maintain the necessary high cathode temperature. The spotless

mode is observed, for example, in so-called super-high-pressure discharge

lamps filled with xenon, which are commercially used in projection devices for

computers. Convective heating can play an important role for the cathode

energy balance. Since ion impact heating is reduced when the spotless mode is

reached, the cathode temperature may fall, reducing thermionic emission. This

may force the cathode to switch back in a mode where only a fraction of the area

is hot, i.e., the thermionic spot mode. The increased ion bombardment heating

may increase the overall cathode temperature, eventually again satisfying the

existence condition of the thermionic spotless mode. Oscillations between ther-

mionic modes have been observed in gas discharge lamps (of course, such lamps

are declared faulty and in need of replacement).Another approach to arc classification is to consider the origin and nature of

the plasma between anode and cathode (Figure 3.41). Of special interest are

Fig. 3.41. Classification of cathode modes based on discharge medium


cases where the plasma contains a large fraction of metal species or is evenexclusively composed of metal, obtained by evaporation of the cathode (‘‘spot-less cathodic vacuum arc’’ [190–192]) or by evaporation of the anode (‘‘anodicarc’’ [193–197]) or by evaporation of cathode material that was deposited on theanode (‘‘refractory anode vacuum arc’’ [198]). All of these modes have theadvantage that metal plasma production does not rely on microscopic explosiveprocesses that are characteristic for cathodic arcs, and hence macroparticlegeneration is avoided or does not play an important role. Deposition rates canbe high, and uniformity issues can be addressed with technological means thatare similar to conventional evaporation techniques. However, the kind ofplasma obtained is different than cathodic arc plasmas in terms of the degreeof ionization, ion charge state distribution, and ion energy. These plasmas arenot further discussed in this book.

Yet other modes of cathodic arcs are obtained at very high arc currents(Iarc441 kA), when the anode is no longer a simple electron collector butvarious ‘‘active’’ anode modes appear. In the world of high-current arcs, typicalfor vacuum arc switches and vacuum arc circuit interrupters, the pathological‘‘low-current’’ case of a passive anode is often called diffuse arc, which must notbe confused with a spotless arc or other modes: diffuse refers here to the anode,while the cathode shows typical cathode spots, which are numerous, small,point-like, and non-stationary. High-current modes include the footpointmode, the anode spot mode, and the intense arc mode, see the reviews by Miller[199, 200].

3.7 The Cohesive Energy Rule

3.7.1 Formulation

Each material has certain characteristics such as ‘‘the ease to burn.’’ This ‘‘ease’’is related to the likelihood that the arc does not spontaneously extinguish; it alsodescribes the level (amplitude) of fluctuations in burning voltage, light emission,ion charge state distribution, ion energies, etc. In this section, these more or lesssubtle differences between cathode materials are discussed in terms of empiricalrules. Among them, the Cohesive Energy Rule [201, 202] appears to be the mostphysically reasonable because it can be associated with fundamental considera-tions of energy conservation and power distribution.

The Cohesive Energy Rule can be formulated as follows: ‘‘The average arcburning voltage of a vacuum arc at a given current is approximately directlyproportional to the cohesive energy of the cathode material.’’

The cohesive energy is the energy needed to form a free, electrically neutralatom by removing it from its bound position in a solid at 0K. Table B8(Appendix B) includes the cohesive energy expressed in eV/atom. It is oftengiven in kJ/mol or kcal/mol or J/g, but it can be expressed in eV/atom, repre-senting the average binding energy of the atom in the solid.

3.7 The Cohesive Energy Rule 149

The arc burning voltage determines the energy dissipated for a given arccurrent, the latter being determined by the electrical discharge circuit. Becausethe dissipated energy affects practically all plasma parameters, the CohesiveEnergy Rule implies a number of secondary rules. This is non-trivial since thecohesive energy is just one of many physical characteristics of a solid, and it isnot obvious why it should be suited to predict the parameters of the expandingvacuum arc plasma. Before this issue is addressed, one may consider otherempirical rules that have been used.

3.7.2 Other Empirical Rules

For decades, researchers have tried, with some success, to identify simple rela-tionships that could help to understand cathode-spot physics and to predictplasma parameters. For example, Kesaev [203] and Grakov [204] attempted tocorrelate the burning voltages with arc current and thermophysical properties ofthe electrode material. Kesaev suggested that there is a correlation between arcvoltage and the product of the boiling temperature and the square root of thethermal conductivity. Nemirovskii and Puchkarev [205] derived a relativelycomplicated relation between burning voltage, the thermal conductivity, andthe specific heat of the cathode material. Brown and co-workers [206, 207] founda correlation between the boiling temperature of the cathodematerial (in Kelvin)and the mean ion charge state of the vacuum arc plasma:

Q ¼ 1þ 3:8� 10�4Tboil: (3:86)

With the large body of experimental data known today it is clear that plasmaparameters such as average ion charge state [206, 208, 209], electron temperature[210, 211], and ion velocity [208, 212] can be correlated to the periodic propertiesof the solid cathode materials – this led to the idea to arrange solid state andplasma properties in Periodic Tables (Appendix B).

3.7.3 Experimental Basis

Measurements of the burning voltage have been done many times. In one of themore recent systematic studies [202], the vacuum arc ion source ‘‘Mevva V’’ [209]was used. The use of this facility had the advantage that the voltage data can bedirectly associated with other data measured at the same facility, such as ioncharge state distributions [207], electron temperatures [211], and directed ionvelocities [212]. Corrections to themeasured voltage data were obtained by usingthe same setup without plasma, i.e., by measuring the voltage for a shortedcathode–anode gap. The lattermeasurements were done not in vacuumbut in airor under inert gas when using reactive cathodematerials. The difference betweenthe voltage measured with plasma and without plasma (shorted gap) is approxi-mately equal to the burning voltage. It is not precisely equal since the shortedgap situation includes the small but non-zero contact resistance of the short. The


error in the measurements increases with increasing current. The contact resis-tance caused the error to be less for noble metals (such as Au, Pt) and higher forvery reactive metals (such Li, Ca, Sr, Ba).

The results of systematic measurements are shown in Figure 3.42 and TableB8 of Appendix B. The experimental data reported in Table B8 are somewhathigher than some literature data because they do not represent the cathode fall orthe minimum voltage needed but rather the voltage obtained by averaging overthe typical noise of the data.

3.7.4 Physical Interpretation

The transition from the cathode’s solid phase to the plasma phase requiresenergy, which is supplied via the power dissipated by the arc,

Parc ¼ V Iarc; (3:87)

whereV is the voltage of the arc (i.e., measured between anode and cathode, andnot to be confused with the voltage provided at the power supply). The energyneeded for the phase transitions is only a fraction of the total energy balance.The total balance of the cathode region has been discussed in Section 3.4.3, andhere it is summarized as

IarcVt ¼ Ephon þ ECE þ Eionization þ Ekin;i þ Eee þ Eth;e þ EMP þ Erad; (3:88)

where t is a time interval over which observation is averaged,Ephon is the phononenergy (heat) transferred to the cathode material, ECE is the cohesive energy

Fig. 3.42. Experimentally found correlation between arc voltage and cohesive energy;

cohesive energy ECE, and arc burning voltage V, for cathode materials of atomicnumber Z. ECE was taken from [213], and the burning voltage was measured at anarc current of 300 A. (From [202])


needed to transfer the cathode material from the solid phase to the vapor phase,Eionization is the energy needed to ionize the vaporized cathode material, Ekin;i is

the kinetic energy given to ions due to the pressure gradient and other accelera-tion mechanisms, Eee is the energy needed to emit electrons from the solid to theplasma (latent work function), Eth;e is the thermal energy (enthalpy) of electrons

in the plasma, EMP is the energy invested in melting, heating, and acceleration ofmacroparticles, and Erad is the energy emitted by radiation. The various terms of(3.88) contribute very differently to the balance, and some of the terms such asenergy invested in macroparticles and radiation are small. The input energy ismostly transferred to heat the cathode, to emit and heat electrons, and toproduce and accelerate ions. The cohesive energy ECE is relatively small andmay be neglected if one wants to calculate the more prominent energy contribu-tions. However, the correlation between burning voltage and cohesive energysuggests to have a closer look at the physical situation.

There are two arguments in the interpretation of the Cohesive Energy Rule,which seems to hold despite the relatively small fraction of energy needed for thephase transition.

The first argument is based on the spatial distribution of the energy input.Figure 3.43 schematically shows that most of the dissipated energy is concen-trated near the cathode surface and associated with the cathode fall. Interest-ingly, driven by the extreme pressure gradient, both electrons and ions areaccelerated away from the cathode surface, carrying away the energy investedin them. Therefore, a very large fraction of input energy is not available toaccomplish the phase transition. Materials with large cohesive energy requiremore energy for the transition from the solid to the vapor phase. The discharge

Fig. 3.43. Schematic illustration of the potential drop near the cathode surface. Thedissipated power density is proportional to the gradient of the potential. The dense

plasma adjacent to the surface receives most of the dissipating power density and istherefore heated rapidly and accelerated away from the cathode, thereby removingmost of the input energy


system can provide this greater energy to the solid by increasing the overallburning voltage. Of course, when this happens, the fraction of energy going toother energy forms, as described by (3.88), increases as well. If this is true, solidswith greater cohesive energy require higher burning voltage, and more energy isalso available for ionization and acceleration of ions, in agreement withmeasurements.

The second interpretation for the Cohesive Energy Rule and its derived rulesis based on the periodicity of many material properties. Periodicity refers here toproperties that are due to the electronic shell structure and that can be groupedaccording to the Periodic Table of the Elements. For example, the meltingtemperature and boiling temperature show the same periodicity as the cohesiveenergy (Tables B1 and B8, Appendix B), which of course is not coincidental butrelated to the electronic structure of atoms and the formation and strength ofchemical bonds in the solid phase. Periodicity is the physical reason that arelation identified for the cohesive energy can be approximately formulated asa relation to the melting or boiling temperature, for example. The question mayrise if a physical relation expressed for the cohesive energy is equivalent to acorresponding relation to the boiling temperature, for example. That is not quitethe case because relations based on the cohesive energy can be associated withthe thermodynamic law of energy conservation. Energy is a physical quantity forwhich one can formulate a balance equation, but temperature is not. Therefore,an empirical rule based on energy should be preferred over a rule related totemperature.

3.7.5 Quantification

In a zero-order approximation one can state that the vacuum arc burningvoltage is about 20V, with somewhat lower values for cathode materials ofsmaller cohesive energy and higher values for materials of greater cohesiveenergy. Quantifying the Cohesive Energy Rule in a first-order approximationgives [214]

V ¼ V0 þ A ECE; (3:89)

where, for the specific experiments with 300-A arcs in a vacuum arc ion source[202], the valuesV0 ¼ 14:3 V andA ¼ 1:69 V=ðeV=atomÞwere found. For otherarc configurations and considering threshold currents rather than 300-A arcs,the constant will be smaller, perhaps as low as V0 � 8 V. Figure 3.44 shows howwell the simple approximation (3.89) represents the Cohesive Energy Rule.Further refinements are discussed in [214].

3.7.6 Related Observations: Ion Erosion and Voltage Noise

The main focus in formulating the Cohesive Energy Rule was the material-dependent arc burning voltage and the underlying energy balance. Other


observations are related such as the metal-dependent ion erosion rate and the

amplitude of voltage fluctuations (noise). It is clear that if more energy is

needed to accomplish the phase transitions from cathodic solid to liquid, gas,

plasma, less material can go through these transitions for a given energy (see

also Section 3.8). This has been experimentally confirmed by carefully studying

the ion erosion rate (Figure 3.45): One can clearly see that cohesive energy and

ion erosion rate are in opposing phase, i.e., the materials of high cohesive

energy show low ion erosion and vice versa.

Fig. 3.44. Comparison of experimental data and Cohesive Energy Rule, approximation(3.89). (From [214])

Fig. 3.45. Ion erosion rate, expressed as ion current normalized by arc current, and

cohesive energy, for most metals of the Periodic Table. One can see that materials of highcohesive energy have low ion erosion rates and vice versa. (From [91])


Another observation that can be related to the Cohesive Energy Rule is theamplitude of voltage noise generated when using different cathode materials.While the Cohesive Energy Rule established an empirical, energy-based rule forthe burning voltage, we now consider the fluctuating component of the burningvoltage. This component can be investigated using fast data acquisitionand Fourier transform analysis [93, 215]. The result was already shown inFigure 3.22. The curves of the spectral power are slightly shifted and theirorder reflects the amplitude of noise: one can see that the higher the cohesiveenergy the greater the voltage noise. Apparently, the higher cohesive energy isagain a suitable measure for the ‘‘difficulty’’ to ignite emission centers.

3.8 Cathode Erosion

The cathode material is the feedstock material for the cathodic arc plasma, andtherefore, as the cathodic arc is burning, the cathode loses mass. Cathodeerosion is actually comprised of three components: material leaving the cathodesurface region as ions, neutral vapor, and macroparticles, i.e.,

�total ¼ �i þ �0 þ �MP: (3:90)

The usually desired form of erosion for cathodic arc coatings is ion erosionbecause ions can be influenced by electric and magnetic fields, and the conden-sation of ions leads to the desirable film properties.

Cathode erosion is typically expressed as mass loss per charge (with the unitmg/As or mg/C). It is most often determined by the weighing method whichimplies that the mass of the cathode is carefully measured before and after arcoperation, and the charge transferred is determined by measuring arc currentand arc duration, i.e.,

�total ¼�mcathodeÐ

Iarcdt: (3:91)

If the arc current is constant, one may simply use

�total ¼�mcathode

Iarctarc: (3:92)

This method does not give any information in which form the loss occurred, i.e.,as ion, neutral, or macroparticle loss.

It should be noted that while this definition of erosion is common, it is notthe best approach from a physical point of view. The normalization of mass lossto charge transferred works relatively well only because the differences inarc voltage are relatively small when the material or discharge conditions arechanged. A better, more fundamental normalization is mass loss per energy, i.e.,

e�total ¼�mcathodeÐIarcVarcdt

: (3:93)

The corresponding unit is mg/J.

3.8 Cathode Erosion 155

One may also define specific erosion rates considering the loss of mass in the

form of ions, neutrals, or macroparticles. To determine the ion erosion rate one

would need to measure the total flux of ions coming from the cathode. This is a

non-trivial task because at least a fraction of ions will condense on the anode.

Weighing the anode is not helpful because macroparticles will also be deposited.

One solution is to measure the ion current to screens, meshes, or shields and to

use geometric correction factors [91].It was already mentioned in Section 3.7.6 that ion erosion rates were found in

opposite phase to the cohesive energy of the cathode material, i.e., the higher the

cohesive energy the lower the ion erosion rate (see Figure 3.45). Because of this

property, the often-quoted ‘‘10%-rule’’ is very approximate.7 The ‘‘10%-rule’’

states that the maximum ion current that can be obtained from a cathodic arc is

about 10% of the arc current. This percentage was identified by erosion studies,

e.g., by Kimblin [216] and Daalder [217]. More recent studies of different

cathode materials [91] confirmed that ion erosion is proportional to the arc

current but they also showed that the percentage scales with the inverse of the

cohesive energy. The percentage can be lower or higher than 10% (Figure 3.46

and Table 3.6).We have the least information on �0, the mass loss by evaporation of neutrals

from the cathode surface. Charge state distribution studies and their interpreta-

tion (e.g., [211]) showed that the plasma generated at cathode spots is fully

ionized, and with very few exceptions (like carbon) one can be sure that the

fraction of neutrals in the spot plasma is very small (less than 1%). However, as

discussed earlier in this chapter, the spot phenomena are fractals in space and

time, and therefore the concept of giving simple percentages for ions and atoms

emitted from the spot region is flawed unless we realize that these are average

values. Only by integrating over time and space we can arrive at those

meaningful average data. They include the vapor from previously active

emission centers, i.e., locations that are not producing plasma anymore but

still hot enough to cause evaporation (stages (iii) and (iv) of the emission site

evolution). We can expect that the vapor pressure of the cathode materials will

play an important role because this quantity can vary over many orders of

magnitude [37].Macroparticle erosion can be the largest fraction of cathodemass loss [219]. It

strongly depends on the material properties, surface conditions, and on the

discharge configuration (like presence of spot-steering magnetic fields). Macro-

particle formation is separately discussed in Chapter 6.

7 Ion erosion is sometimes expressed as ion current normalized by the arc current, giving

the unit ‘‘%.’’ Contrary to popular reading, this does not mean that ions contribute x%to the arc current, and electrons (100–x)%. Because the measured ion flux moves awayfrom the cathode, electrons need to carry more than 100% of the arc current, compen-sating for the current of ions going in the ‘‘wrong direction.’’


Fig. 3.46. Ion current from a cathodic arc discharge, as a function of arc current, with thematerial as a parameter. (After [91])

Table 3.6. Ion current normalized by arc current,i, and ion erosion rates, �i, for various

cathode materials. The last column shows the ion erosion rate normalized by arc energy,~�i, calculated from �i and the arc voltage given in Table B8

Cathode

meterial

Reference

[216]

Reference

[218]

Reference

[42]

Reference

[91]

Reference

[91]

i (%) i (%) �i (mg/C) i (%) �i (mg/C) ~�i (mg/J)

C 10.0 19 13–17 19 23.8 0.804

Mg – 12.7 19–25 12.7 18.8 1.00Al – 11.2 22–25 11.2 15.9 0.674Ti 8.0 9.7 – 9.7 22.4 1.05

Co 8.0 9.6 – 9.6 30.4 1.33Cu – 11.4 35–39 11.4 33.4 1.42Zr – 10.5 – 10.5 36.3 1.55Cd 8.0 12 128–130 12 94.6 5.90

In – 10.2 – 10.2 80.5 4.60Sn – 11.4 – 11.4 83.1 4.75Sm – 6.5 – 6.5 46.1 3.16

Ta – 5.3 – 5.3 31.2 1.09W 7.0 5 62–90 5 27.1 0.850Pt – 5.6 – 5.6 50.6 2.25

Pb – 14.3 – 14.3 172.8 11.1Bi – 10.2 – 10.2 171.5 11.0

3.8 Cathode Erosion 157

3.9 Plasma Formation

3.9.1 Phase Transitions

Let us have a closer look at the phase transformations that lead to the three kindsof cathode erosion (ions, neutrals, macroparticles). In particular we will considerhow the cathodic arc plasma is formed by studying the path of the material inphase diagrams. With few exceptions, such as mercury which is already in the

liquid phase at room temperature, it is clear that the cathode processes start withthe cathode material in its solid phase. Due to the high energy density of thecathode-spot region, the cathode material goes through all four phases: solid!liquid! vapor! plasma.

One may plot a density–temperature phase diagram containing all phases,phase boundaries, and critical points, should they exist. For the axes of the phasediagram one could select temperature and density of heavy particles or electrons. Itis acceptable to speak of one temperature, as opposed to electron temperature orion temperature, provided collisions are sufficiently frequent. The phase diagram,

Figure 3.47, contains regions of solid, liquid, and gas (vapor) as well as coexistenceregions for solid–liquid, liquid–gas, and solid–gas. It also shows a critical point ‘‘C’’which is the point of highest temperature where liquid and gas are still in equili-brium. BeyondC, one cannot distinguish between liquid and vapor. In the example

Fig. 3.47. Equilibrium phase diagram for copper using temperature–electron density

presentation. One sees the classical phases as well as plasma at high temperature andlow density. The two straight lines indicate degeneration of electrons and borderlineof region with strong non-ideal coupling. The important information is in the path of

cathode material. (Adapted from [45] and [211])


shown in Figure 3.47, copper is selected because many data are available (copperbeing the ‘‘guinea pig’’ material of arc physicists and electrical contact engineers).The figure shows two out of a large number of possible paths of the material whentransitioning from solid to expanded plasma [45].

In the explosive stage of an emission site, comparable to a wire explosion, thematerial is heated so rapidly that its density initially remains almost constant.Therefore, the path in the diagram goes to the right almost horizontally. It isinteresting to note that the temperature of the small volume under considerationcan exceed the temperature of the critical point C, and hence the material can bea supercritical fluid that gradually transitions into a fully ionized plasma withoutever having gone through the classical liquid and vapor phases. This transition isextremely fast, shorter than 1 ns, and likely to occur only at the beginning of theexplosive stage of a new emission site. The material reaches a maximum tem-perature and, driven by the extreme pressure gradient to the surroundingvacuum or low-pressure plasma, it expands and cools down. Eventually, aftersome 10 ns, the plasma is not in equilibrium anymore and one needs to distin-guish between the temperatures of electrons and ions.

In the later development stages of the emission site, processes are less violent,and the path is closer to what one would normally expect: the cathode material isstill rapidly heated but expands slightly, melts, vaporizes, and becomes ionizedin the vicinity of the spot. The plasma in the expanded plasma is different thanthe plasma produced from the supercritical fluid, which contributes to the rapidfluctuations seen in cathodic arc plasmas. In this later stage, the material is lessionized, and cathode erosion may become dominated by evaporation where thenecessary energy is provided by ion bombardment from the plasma [220].

Due to the generally explosive nature of cathode processes, the path ofmaterial in the phase diagram is not constant but rapidly changing betweenthe extremes located on both sides of the critical point.

3.9.2 Non-ideal Plasma

When the cathode material takes the path of explosive transformation fromsolid to plasma, circumnavigating the critical point (Figure 3.47), there is acertain, short-lived, high-density state that is best described as non-ideal plasma.For comparison, in the more common, ideal plasma, the kinetic energy ofplasma particles is much greater than the interaction energy, which is mainlydue to shielded Coulomb interaction. Equivalently, one can say that there aremany charged particles in the Debye sphere, i.e., a sphere whose radius is thelocal Debye length. These properties allowed us to introduce the approximationof binary collisions between charged particles. In non-ideal plasmas, in contrast,the density is very high, and the interaction between charged particles is muchstronger. The binary collision approximation is no longer a good approximationbut one needs to consider multi-particle interactions, especially when the plasmadensity approaches solid-state density.

3.9 Plasma Formation 159

There are many texts on non-ideal plasmas [221–223], and here it should

suffice to give the parameters that characterize the degree of coupling or non-

ideality. A plasma is called non-ideal or strongly coupled if the potential energy

of the plasma particles due to Coulomb interaction is not much smaller than

the average kinetic energy. To quantify this, a dimensionless coupling parameter

�e can be defined as the ratio of potential to kinetic energy:

�e ¼e2

4p"0kT4pne3

� �1=3

; (3:94)

where the symbols have the usual meaning, like "0 is the permittivity of free space

and k is the Boltzmann constant. The border between classical andquantum regions

for the electron gas can be identified by the dimensionless degeneration parameter

�e ¼ ne�3e ; (3:95)

where ne is the electron density and �e is a characteristic quantum length, the

electron de Broglie wavelength:

�e ¼h

2pmekTð Þ1=2: (3:96)

One of the features of non-ideal plasmas is the enhancement of the degree of

ionization by pressure ionization, a multi-particle interaction effect observed at

very high particle densities and pressure, as it exists during the explosive stage of

an emission site [45]. The Coulomb interaction of the outer bound electrons of

an atom or ion with the surrounding charged particles (ions and free electrons)

and with polarizable particles (atoms and clusters) leads to a substantial low-

ering of their binding energy. For densities approaching solid-state density, this

lowering is additionally strengthened by the quantum-mechanical exchange

interaction between the bound-shell electrons of neighboring particles. With

increasing density, the outer shells (bound electron states) become compressed

and finally disappear. The mean ionization state continuously increases and is

much higher in real plasmas at high pressure than one would expect on the basis

of the more conventional theories which neglect such interactions. In the asymp-

totic limit of very high density, the high-pressure plasma is fully ionized, con-

sisting only of free, bare nuclei and free electrons, whereas the ionization of an

ideal model plasma (i.e., where interactions are neglected) tends to zero, with

finally all charges bound in neutral atoms (Figure 3.48).The cathode material in the explosive stage of an emission site follows a path

from high to low density, which corresponds going from right to left in

Figure 3.48 (putting aside, for a moment, that the temperature is not constant

as assumed in that figure). The material transitions through a ‘‘valley of low

ionization’’ for moderately non-ideal plasma conditions. This valley can be

associated with a region where the conductivity is lowest (resistivity highest)

and therefore power dissipation is highest. This can be understood taking into

account that on the high-density side (right side of Figure 3.48), electrons


degenerate, form conduction bands, and follow Fermi–Dirac statistics with aFermi distribution (Section 3.2.1), hence there is good metallic conduction. Onthe low-density side, electrons are not degenerated and can therefore bedescribed by Boltzmann statistics leading to a Maxwell distribution, and wehave the high conductivity of a fully ionized plasma (the so-called Spitzerconductivity [224]). In the transition, the Fermi distribution asymptoticallyapproaches the Boltzmann distribution. The now-free electrons in the denseplasma suffer many collisions due to the high density, which appear like ‘‘fric-tion’’ to the transport of charge, hence the conductivity is reduced.

As the dense plasma expands, it moves through the region in the phasediagram where the curves of non-ideal and ideal calculation merge: this is theregion of weakly non-ideal plasma. With further decreasing density, the fre-quency of collisions falls and therefore the condition for equilibrium are even-tually violated: the plasma transitions into non-equilibrium, which plays animportant role when considering plasma properties in the interelectrode space(Chapter 4).

One shortcoming of using Figure 3.48 is that it shows curves for constanttemperature. In the explosive phase transition, matter is first rapidly heatedfollowed by cooling during expansion. One could imagine a set of isothermalcurves, each showing a valley of low ionization. The path of the plasma wouldintersect a number of those isothermals. Qualitatively, the physical pictureremains the same.

Fig. 3.48. Equilibrium calculations for ideal and non-ideal copper plasma at densitiesapproaching solid state: for the ideal model plasma, the lowering of the ionizationenergy by multi-particle interaction was neglected, leading to unrealistically low

ionization. The actual ionization curve describes a ‘‘valley of low ionization’’ in theregion of weakly non-ideal parameters. (After [45])

3.9 Plasma Formation 161

Transient non-ideal plasma on the one hand and ‘‘holes’’ in the cathodesheath (Figure 3.13) on the other hand are intimately connected. The cathodemetal transitions from solid to fully ionized plasma, i.e., circumnavigating thecritical point, which requires that very high plasma densities exist. Both electronsand ions are present and can contribute to transport of the discharge current;however, the plasma is in the ‘‘valley of low ionization,’’ as explained before.Under these transient conditions, which are spatially limited, the conductivity ofthe plasma is relatively low, and the voltage drop, which is normally associatedwith the cathode sheath, is in the non-ideal plasma, as opposed to that in thesheath. Cathode models throughout the literature focus on electron emissionfrom a cathode surface, and various surface and emission conditions areassumed. The transient formation of holes in the sheath and the phase transitionon the high-density side of the critical point are usually not considered. Therelevance of this mechanism is still subject to further experimental diagnosticsand theoretical modeling.

3.9.3 Ion Acceleration

This chapter is concluded by briefly considering ion acceleration near cathodespots because this will naturally lead to the next chapter in which the interelec-trode plasma is discussed. In this section, only those acceleration processes areconsidered that are intimately related to plasma generation and electrodeprocesses.

Ions ‘‘born’’ at cathode spots are known to have high velocity once they areobserved ormeasured in distances that are large compared to the size of the spot.In fact, ion velocities are supersonic with respect to the ion sound velocity. Untilnow, there is no generally accepted theory of ion acceleration, which is in partdue to the lack of reliable data on charge- and time-dependent velocity distribu-tion functions. Although great progress was made, there are still questions onhow much the electric field is contributing to ion acceleration. Another possible(or actually most likely and important) reason is that ion acceleration theoriesgenerally do not consider the very different stages of the emission site’s life cycle.It is safe to assume that ions generated in the explosive stage experience differentaccelerating forces than ions generated in later stages. Due to the existence ofnon-stationary processes, one should a priori expect non-stationary velocitydistribution functions. So far, only average velocities or kinetic energies orvelocity distribution functions have been measured, although all researchersstress the great variability of velocity data.

Ion acceleration near emission sites (cathode spots or spot fragments) arethought to be due to the following forces: (i) pressure gradient of ions, (ii) pressuregradient of electrons, (iii) collective acceleration of ions by electron–ion coup-ling, and (iv) acceleration by an electric field if such field exists (potential hump).Explanation and theoretical modeling have been challenging and the subject ofmany papers over the years [114, 225–231].


The fractal model with explosive emission events as the short-time cutoff isbest suited to (at least qualitatively) explain a range of experimental results,especially those obtained with high resolution, which show a wide spectrum offluctuations. However, other models are also conceivable, and more tractable,by not considering the fluctuating features but rather by trying to explain thegeneral, averaged trends.

In the simplest case one would use a stationary model in one dimension,keeping in mind that the results cannot explain any fluctuation or spatialvariation other than on axis. Considering ions and electrons as two fluids ledto a set of two-fluid equations that exhibit a saddle point at the ion soundvelocity [227]. Transition to transonic flow velocities (i.e., ions become super-sonic) is only possible through the directed friction force associated with elec-tron–ion collisions. A small potential hump has only a minor effect for themomentum gain of the ions [226, 227].

Taking themultifluid equations that describe anisotropic plasma expansion inspherical coordinates, Hantzsche [229] found an analytical solution in the formof asymptotic power series. According to this model, the main force responsiblefor the high ion velocities is electron–ion friction, i.e., electrons movemuch fasterand with greater quantities from the emission center; they transfer kinetic energyto the slower ions via Coulomb interaction. This mechanism provides roughlyhalf of the kinetic energy given to the ions. The electrons, in turn, are acceleratedby the electric field and the electron pressure gradient. The remaining ion accel-eration is caused by the ion pressure gradient and the electric field force. Overall,most of the acceleration is completed in a space less than 10 mm from the emissioncenter, and the velocities are approximately constant for distances greater than100 mm. Those average ion velocities are further discussed in Chapter 4.

Other models are conceivable, too. For example, considering a mercuryvacuum arc, Beilis [232] assumed that the near-cathode region consists of threelayered regions, which – in the order of their distance from the cathode surface –can be called the first plasma region, the double sheath, and the second plasmaregion. Electrons emitted from the surface of the first plasma region are acceler-ated through the double sheath region (potential drop 10–15V) into the secondplasma region. Likewise ions flowing from the second ion region are acceleratedthrough the double sheath into the first plasma region where they serve as asignificant heat source. Two types of time-dependent solutions with character-istic times of 0.1–1 and 100 ms exist [232]. However, this model is difficult to bringinto relation to the fractal structure of emission sites advocated in this chapter.

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